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{{task}}
An [[wp:Abundant_number|Abundant number]] is a number '''n''' for which the ''sum of divisors'' '''σ(n) > 2n''',
or, equivalently, the ''sum of proper divisors'' (or aliquot sum) '''s(n) > n'''.
;E.G.: '''12''' is abundant, it has the proper divisors '''1,2,3,4 & 6''' which sum to '''16''' ( > '''12''' or '''n''');
or alternately, has the sigma sum of '''1,2,3,4,6 <small>&</small> 12''' which sum to '''28''' ( > '''24''' or '''2n''').
Abundant numbers are common, though '''even''' abundant numbers seem to be much more common than '''odd''' abundant numbers.
To make things more interesting, this task is specifically about finding ''odd abundant numbers''.
;Task *Find and display here: at least the first 25 abundant odd numbers and either their proper divisor sum or sigma sum. *Find and display here: the one thousandth abundant odd number and either its proper divisor sum or sigma sum. *Find and display here: the first abundant odd number greater than one billion (109) and either its proper divisor sum or sigma sum.
;References: :* the OEIS entry: [http://oeis.org/A005231 odd abundant numbers (odd numbers n whose sum of divisors exceeds 2n)]. :* American Journal of Mathematics, Vol. 35, No. 4 (Oct., 1913), pp. 413-422 - Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors (LE Dickson)
360 Assembly
* Abundant odd numbers 18/09/2019
ABUNODDS CSECT
USING ABUNODDS,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
SAVE (14,12) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R8,0 n=0
LA R6,3 i=3
DO WHILE=(C,R8,LT,NN1) do i=3 by 2 until n>=nn1
BAL R14,SIGMA s=sigma(i)
IF CR,R9,GT,R6 THEN if s>i then
LA R8,1(R8) n++
BAL R14,PRINT print results
ENDIF , endif
LA R6,2(R6) i+=2
ENDDO , enddo i
LA R8,0 n=0
LA R6,3 i=3
XR R1,R1 f=false
DO WHILE=(C,R1,EQ,=F'0') do i=3 by 2 while not f
BAL R14,SIGMA s=sigma(i)
IF CR,R9,GT,R6 THEN if s>i then
LA R8,1(R8) n++
IF C,R8,GE,NN2 THEN if n>=nn2 then
BAL R14,PRINT print results
LA R1,1 f=true
ENDIF , endif
ENDIF , endif
LA R6,2(R6) i+=2
ENDDO , enddo i
LA R8,0 n=0
L R6,NN3 i=mm3
LA R6,1(R6) +1
XR R1,R1 f=false
DO WHILE=(C,R1,EQ,=F'0') do i=nn3+1 by 2 while not f
BAL R14,SIGMA s=sigma(i)
IF CR,R9,GT,R6 THEN if s>i then
BAL R14,PRINT print results
LA R1,1 f=true
ENDIF , endif
LA R6,2(R6) i+=2
ENDDO , enddo i
L R13,4(0,R13) restore previous savearea pointer
RETURN (14,12),RC=0 restore registers from calling save
SIGMA CNOP 0,4 ---- subroutine sigma
LA R9,1 s=1
LA R7,3 j=3
LR R5,R7 j
MR R4,R7 j*j
DO WHILE=(CR,R5,LT,R6) do j=3 by 2 while j*j<i
LR R4,R6 i
SRDA R4,32 ~
DR R4,R7 i/j
IF LTR,R4,Z,R4 THEN if mod(i,j)=0 then
AR R9,R7 s+j
LR R4,R6 i
SRDA R4,32 ~
DR R4,R7 i/j
AR R9,R5 s=s+j+i/j
ENDIF , endif
LA R7,2(R7) j+=2
LR R5,R7 j
MR R4,R7 j*j
ENDDO , enddo j
IF CR,R5,EQ,R6 THEN if j*j=i then
AR R9,R7 s=s+j
ENDIF , endif
BR R14 ---- end of subroutine sigma
PRINT CNOP 0,4 ---- subroutine print
XDECO R8,XDEC edit n
MVC BUF(4),XDEC+8 output n
XDECO R6,BUF+14 edit & output i
XDECO R9,BUF+33 edit & output s
XPRNT BUF,L'BUF print buffer
BR R14 ---- end of subroutine print
NN1 DC F'25' nn1=25
NN2 DC F'1000' nn2=1000
NN3 DC F'1000000000' nn3=1000000000
BUF DC CL80'.... - number=............ sigma=............'
XDEC DS CL12 temp for edit
REGEQU equate registers
END ABUNODDS
{{out}}
1 - number= 945 sigma= 975
2 - number= 1575 sigma= 1649
3 - number= 2205 sigma= 2241
4 - number= 2835 sigma= 2973
5 - number= 3465 sigma= 4023
6 - number= 4095 sigma= 4641
7 - number= 4725 sigma= 5195
8 - number= 5355 sigma= 5877
9 - number= 5775 sigma= 6129
10 - number= 5985 sigma= 6495
11 - number= 6435 sigma= 6669
12 - number= 6615 sigma= 7065
13 - number= 6825 sigma= 7063
14 - number= 7245 sigma= 7731
15 - number= 7425 sigma= 7455
16 - number= 7875 sigma= 8349
17 - number= 8085 sigma= 8331
18 - number= 8415 sigma= 8433
19 - number= 8505 sigma= 8967
20 - number= 8925 sigma= 8931
21 - number= 9135 sigma= 9585
22 - number= 9555 sigma= 9597
23 - number= 9765 sigma= 10203
24 - number= 10395 sigma= 12645
25 - number= 11025 sigma= 11946
1000 - number= 492975 sigma= 519361
0 - number= 1000000575 sigma= 1083561009
Ada
This solution uses the package ''Generic_Divisors'' from the Proper Divisors task [[http://rosettacode.org/wiki/Proper_divisors#Ada]].
with Ada.Text_IO, Generic_Divisors;
procedure Odd_Abundant is
function Same(P: Positive) return Positive is (P);
package Divisor_Sum is new Generic_Divisors
(Result_Type => Natural, None => 0, One => Same, Add => "+");
function Abundant(N: Positive) return Boolean is
(Divisor_Sum.Process(N) > N);
package NIO is new Ada.Text_IO.Integer_IO(Natural);
Current: Positive := 1;
procedure Print_Abundant_Line
(Idx: Positive; N: Positive; With_Idx: Boolean:= True) is
begin
if With_Idx then
NIO.Put(Idx, 6); Ada.Text_IO.Put(" |");
else
Ada.Text_IO.Put(" *** |");
end if;
NIO.Put(N, 12); Ada.Text_IO.Put(" | ");
NIO.Put(Divisor_Sum.Process(N), 12); Ada.Text_IO.New_Line;
end Print_Abundant_Line;
begin
-- the first 25 abundant odd numbers
Ada.Text_IO.Put_Line(" index | number | proper divisor sum ");
Ada.Text_IO.Put_Line("-------+-------------+--------------------");
for I in 1 .. 25 loop
while not Abundant(Current) loop
Current := Current + 2;
end loop;
Print_Abundant_Line(I, Current);
Current := Current + 2;
end loop;
-- the one thousandth abundant odd number
Ada.Text_IO.Put_Line("-------+-------------+--------------------");
for I in 26 .. 1_000 loop
Current := Current + 2;
while not Abundant(Current) loop
Current := Current + 2;
end loop;
end loop;
Print_Abundant_Line(1000, Current);
-- the first abundant odd number greater than 10**9
Ada.Text_IO.Put_Line("-------+-------------+--------------------");
Current := 10**9+1;
while not Abundant(Current) loop
Current := Current + 2;
end loop;
Print_Abundant_Line(1, Current, False);
end Odd_Abundant;
{{out}}
Index | Number | proper divisor sum
-------+-------------+--------------------
1 | 945 | 975
2 | 1575 | 1649
3 | 2205 | 2241
4 | 2835 | 2973
5 | 3465 | 4023
6 | 4095 | 4641
7 | 4725 | 5195
8 | 5355 | 5877
9 | 5775 | 6129
10 | 5985 | 6495
11 | 6435 | 6669
12 | 6615 | 7065
13 | 6825 | 7063
14 | 7245 | 7731
15 | 7425 | 7455
16 | 7875 | 8349
17 | 8085 | 8331
18 | 8415 | 8433
19 | 8505 | 8967
20 | 8925 | 8931
21 | 9135 | 9585
22 | 9555 | 9597
23 | 9765 | 10203
24 | 10395 | 12645
25 | 11025 | 11946
-------+-------------+--------------------
1000 | 492975 | 519361
-------+-------------+--------------------
*** | 1000000575 | 1083561009
ALGOL 68
BEGIN
# find some abundant odd numbers - numbers where the sum of the proper #
# divisors is bigger than the number #
# itself #
# returns the sum of the proper divisors of n #
PROC divisor sum = ( INT n )INT:
BEGIN
INT sum := 1;
FOR d FROM 2 TO ENTIER sqrt( n ) DO
IF n MOD d = 0 THEN
sum +:= d;
IF INT other d := n OVER d;
other d /= d
THEN
sum +:= other d
FI
FI
OD;
sum
END # divisor sum # ;
# find numbers required by the task #
BEGIN
# first 25 odd abundant numbers #
INT odd number := 1;
INT a count := 0;
INT d sum := 0;
print( ( "The first 25 abundant odd numbers:", newline ) );
WHILE a count < 25 DO
IF ( d sum := divisor sum( odd number ) ) > odd number THEN
a count +:= 1;
print( ( whole( odd number, -6 )
, " proper divisor sum: "
, whole( d sum, 0 )
, newline
)
)
FI;
odd number +:= 2
OD;
# 1000th odd abundant number #
WHILE a count < 1 000 DO
IF ( d sum := divisor sum( odd number ) ) > odd number THEN
a count := a count + 1
FI;
odd number +:= 2
OD;
print( ( "1000th abundant odd number:"
, newline
, " "
, whole( odd number - 2, 0 )
, " proper divisor sum: "
, whole( d sum, 0 )
, newline
)
);
# first odd abundant number > one billion #
odd number := 1 000 000 001;
BOOL found := FALSE;
WHILE NOT found DO
IF ( d sum := divisor sum( odd number ) ) > odd number THEN
found := TRUE;
print( ( "First abundant odd number > 1 000 000 000:"
, newline
, " "
, whole( odd number, 0 )
, " proper divisor sum: "
, whole( d sum, 0 )
, newline
)
)
FI;
odd number +:= 2
OD
END
END
{{out}}
The first 25 abundant odd numbers:
945 proper divisor sum: 975
1575 proper divisor sum: 1649
2205 proper divisor sum: 2241
2835 proper divisor sum: 2973
3465 proper divisor sum: 4023
4095 proper divisor sum: 4641
4725 proper divisor sum: 5195
5355 proper divisor sum: 5877
5775 proper divisor sum: 6129
5985 proper divisor sum: 6495
6435 proper divisor sum: 6669
6615 proper divisor sum: 7065
6825 proper divisor sum: 7063
7245 proper divisor sum: 7731
7425 proper divisor sum: 7455
7875 proper divisor sum: 8349
8085 proper divisor sum: 8331
8415 proper divisor sum: 8433
8505 proper divisor sum: 8967
8925 proper divisor sum: 8931
9135 proper divisor sum: 9585
9555 proper divisor sum: 9597
9765 proper divisor sum: 10203
10395 proper divisor sum: 12645
11025 proper divisor sum: 11946
1000th abundant odd number:
492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
1000000575 proper divisor sum: 1083561009
BASIC256
{{trans|Visual Basic .NET}}
numimpar = 1
contar = 0
sumaDiv = 0
function SumaDivisores(n)
# Devuelve la suma de los divisores propios de n
suma = 1
i = int(sqr(n))
for d = 2 to i
if n % d = 0 then
suma += d
otroD = n \ d
if otroD <> d Then suma += otroD
end if
Next d
Return suma
End Function
# Encontrar los números requeridos por la tarea:
# primeros 25 números abundantes impares
Print "Los primeros 25 números impares abundantes:"
While contar < 25
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then
contar += 1
Print numimpar & " suma divisoria adecuada: " & sumaDiv
End If
numimpar += 2
End While
# 1000er número impar abundante
While contar < 1000
sumaDiv = SumaDivisores(numimpar)
print sumaDiv & " " & contar
If sumaDiv > numimpar Then contar += 1
numimpar += 2
End While
Print Chr(10) & "1000º número impar abundante:"
Print " " & (numimpar - 2) & " suma divisoria adecuada: " & sumaDiv
# primer número impar abundante > mil millones (millardo)
numimpar = 1000000001
encontrado = False
While Not encontrado
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then
encontrado = True
Print Chr(10) & "Primer número impar abundante > 1 000 000 000:"
Print " " & numimpar & " suma divisoria adecuada: " & sumaDiv
End If
numimpar += 2
End While
End
C
#include <stdio.h> #include <math.h> // The following function is for odd numbers ONLY // Please use "for (unsigned i = 2, j; i*i <= n; i ++)" for even and odd numbers unsigned sum_proper_divisors(const unsigned n) { unsigned sum = 1; for (unsigned i = 3, j; i < sqrt(n)+1; i += 2) if (n % i == 0) sum += i + (i == (j = n / i) ? 0 : j); return sum; } int main(int argc, char const *argv[]) { unsigned n, c; for (n = 1, c = 0; c < 25; n += 2) if (n < sum_proper_divisors(n)) printf("%u: %u\n", ++c, n); for ( ; c < 1000; n += 2) if (n < sum_proper_divisors(n)) c ++; printf("\nThe one thousandth abundant odd number is: %u\n", n); for (n = 1000000001 ;; n += 2) if (n < sum_proper_divisors(n)) break; printf("The first abundant odd number above one billion is: %u\n", n); return 0; }
{{out}}
1: 945
2: 1575
3: 2205
4: 2835
5: 3465
6: 4095
7: 4725
8: 5355
9: 5775
10: 5985
11: 6435
12: 6615
13: 6825
14: 7245
15: 7425
16: 7875
17: 8085
18: 8415
19: 8505
20: 8925
21: 9135
22: 9555
23: 9765
24: 10395
25: 11025
The one thousandth abundant odd number is: 492977
The first abundant odd number above one billion is: 1000000575
C++
{{trans|Go}}
#include <algorithm> #include <iostream> #include <numeric> #include <sstream> #include <vector> std::vector<int> divisors(int n) { std::vector<int> divs{ 1 }; std::vector<int> divs2; for (int i = 2; i*i <= n; i++) { if (n%i == 0) { int j = n / i; divs.push_back(i); if (i != j) { divs2.push_back(j); } } } std::copy(divs2.crbegin(), divs2.crend(), std::back_inserter(divs)); return divs; } int sum(const std::vector<int>& divs) { return std::accumulate(divs.cbegin(), divs.cend(), 0); } std::string sumStr(const std::vector<int>& divs) { auto it = divs.cbegin(); auto end = divs.cend(); std::stringstream ss; if (it != end) { ss << *it; it = std::next(it); } while (it != end) { ss << " + " << *it; it = std::next(it); } return ss.str(); } int abundantOdd(int searchFrom, int countFrom, int countTo, bool printOne) { int count = countFrom; int n = searchFrom; for (; count < countTo; n += 2) { auto divs = divisors(n); int tot = sum(divs); if (tot > n) { count++; if (printOne && count < countTo) { continue; } auto s = sumStr(divs); if (printOne) { printf("%d < %s = %d\n", n, s.c_str(), tot); } else { printf("%2d. %5d < %s = %d\n", count, n, s.c_str(), tot); } } } return n; } int main() { using namespace std; const int max = 25; cout << "The first " << max << " abundant odd numbers are:\n"; int n = abundantOdd(1, 0, 25, false); cout << "\nThe one thousandth abundant odd number is:\n"; abundantOdd(n, 25, 1000, true); cout << "\nThe first abundant odd number above one billion is:\n"; abundantOdd(1e9 + 1, 0, 1, true); return 0; }
{{out}}
The first 25 abundant odd numbers are:
1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
C#
using static System.Console; using System.Collections.Generic; using System.Linq; public static class AbundantOddNumbers { public static void Main() { WriteLine("First 25 abundant odd numbers:"); foreach (var x in AbundantNumbers().Take(25)) WriteLine(x.Format()); WriteLine(); WriteLine($"The 1000th abundant odd number: {AbundantNumbers().ElementAt(999).Format()}"); WriteLine(); WriteLine($"First abundant odd number > 1b: {AbundantNumbers(1_000_000_001).First().Format()}"); } static IEnumerable<(int n, int sum)> AbundantNumbers(int start = 3) => start.UpBy(2).Select(n => (n, sum: n.DivisorSum())).Where(x => x.sum > x.n); static int DivisorSum(this int n) => 3.UpBy(2).TakeWhile(i => i * i <= n).Where(i => n % i == 0) .Select(i => (a:i, b:n/i)).Sum(p => p.a == p.b ? p.a : p.a + p.b) + 1; static IEnumerable<int> UpBy(this int n, int step) { for (int i = n; ; i+=step) yield return i; } static string Format(this (int n, int sum) pair) => $"{pair.n:N0} with sum {pair.sum:N0}"; }
{{out}}
First 25 abundant odd numbers:
945 with sum 975
1,575 with sum 1,649
2,205 with sum 2,241
2,835 with sum 2,973
3,465 with sum 4,023
4,095 with sum 4,641
4,725 with sum 5,195
5,355 with sum 5,877
5,775 with sum 6,129
5,985 with sum 6,495
6,435 with sum 6,669
6,615 with sum 7,065
6,825 with sum 7,063
7,245 with sum 7,731
7,425 with sum 7,455
7,875 with sum 8,349
8,085 with sum 8,331
8,415 with sum 8,433
8,505 with sum 8,967
8,925 with sum 8,931
9,135 with sum 9,585
9,555 with sum 9,597
9,765 with sum 10,203
10,395 with sum 12,645
11,025 with sum 11,946
The 1000th abundant odd number: 492,975 with sum 519,361
First abundant odd number > 1b: 1,000,000,575 with sum 1,083,561,009
D
{{trans|C++}}
import std.stdio; int[] divisors(int n) { import std.range; int[] divs = [1]; int[] divs2; for (int i = 2; i * i <= n; i++) { if (n % i == 0) { int j = n / i; divs ~= i; if (i != j) { divs2 ~= j; } } } divs ~= retro(divs2).array; return divs; } int abundantOdd(int searchFrom, int countFrom, int countTo, bool printOne) { import std.algorithm.iteration; import std.array; import std.conv; int count = countFrom; int n = searchFrom; for (; count < countTo; n += 2) { auto divs = divisors(n); int tot = sum(divs); if (tot > n) { count++; if (printOne && count < countTo) { continue; } auto s = divs.map!(to!string).join(" + "); if (printOne) { writefln("%d < %s = %d", n, s, tot); } else { writefln("%2d. %5d < %s = %d", count, n, s, tot); } } } return n; } void main() { const int max = 25; writefln("The first %d abundant odd numbers are:", max); int n = abundantOdd(1, 0, 25, false); writeln("\nThe one thousandth abundant odd number is:"); abundantOdd(n, 25, 1000, true); writeln("\nThe first abundant odd number above one billion is:"); abundantOdd(cast(int)(1e9 + 1), 0, 1, true); }
{{out}}
The first 25 abundant odd numbers are:
1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Factor
USING: arrays formatting io kernel lists lists.lazy math
math.primes.factors sequences tools.memory.private ;
IN: rosetta-code.abundant-odd-numbers
: σ ( n -- sum ) divisors sum ;
: abundant? ( n -- ? ) [ σ ] [ 2 * ] bi > ;
: abundant-odds-from ( n -- list )
dup even? [ 1 + ] when
[ 2 + ] lfrom-by [ abundant? ] lfilter ;
: first25 ( -- seq ) 25 1 abundant-odds-from ltake list>array ;
: 1,000th ( -- n ) 1 abundant-odds-from 999 [ cdr ] times car ;
: first>10^9 ( -- n ) 1,000,000,001 abundant-odds-from car ;
GENERIC: show ( obj -- )
M: integer show dup σ [ commas ] bi@ "%-6s σ = %s\n" printf ;
M: array show [ show ] each ;
: abundant-odd-numbers-demo ( -- )
first25 "First 25 abundant odd numbers:"
1,000th "1,000th abundant odd number:"
first>10^9 "First abundant odd number > one billion:"
[ print show nl ] 2tri@ ;
MAIN: abundant-odd-numbers-demo
{{out}}
First 25 abundant odd numbers:
945 σ = 1,920
1,575 σ = 3,224
2,205 σ = 4,446
2,835 σ = 5,808
3,465 σ = 7,488
4,095 σ = 8,736
4,725 σ = 9,920
5,355 σ = 11,232
5,775 σ = 11,904
5,985 σ = 12,480
6,435 σ = 13,104
6,615 σ = 13,680
6,825 σ = 13,888
7,245 σ = 14,976
7,425 σ = 14,880
7,875 σ = 16,224
8,085 σ = 16,416
8,415 σ = 16,848
8,505 σ = 17,472
8,925 σ = 17,856
9,135 σ = 18,720
9,555 σ = 19,152
9,765 σ = 19,968
10,395 σ = 23,040
11,025 σ = 22,971
1,000th abundant odd number:
492,975 σ = 1,012,336
First abundant odd number > one billion:
1,000,000,575 σ = 2,083,561,584
FreeBASIC
{{trans|Visual Basic .NET}}
Declare Function SumaDivisores(n As Integer) As Integer
Dim numimpar As Integer = 1
Dim contar As Integer = 0
Dim sumaDiv As Integer = 0
Function SumaDivisores(n As Integer) As Integer
' Devuelve la suma de los divisores propios de n
Dim suma As Integer = 1
Dim As Integer d, otroD
For d = 2 To Cint(Sqr(n))
If n Mod d = 0 Then
suma += d
otroD = n \ d
If otroD <> d Then suma += otroD
End If
Next d
Return suma
End Function
' Encontrar los números requeridos por la tarea:
' primeros 25 números abundantes impares
Print "Los primeros 25 números impares abundantes:"
Do While contar < 25
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then
contar += 1
Print using "######"; numimpar;
Print " suma divisoria adecuada: " & sumaDiv
End If
numimpar += 2
Loop
' 1000er número impar abundante
Do While contar < 1000
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then contar += 1
numimpar += 2
Loop
Print Chr(10) & "1000º número impar abundante:"
Print " " & (numimpar - 2) & " suma divisoria adecuada: " & sumaDiv
' primer número impar abundante > mil millones (millardo)
numimpar = 1000000001
Dim encontrado As Boolean = False
Do While Not encontrado
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then
encontrado = True
Print Chr(10) & "Primer número impar abundante > 1 000 000 000:"
Print " " & numimpar & " suma divisoria adecuada: " & sumaDiv
End If
numimpar += 2
Loop
End
{{out}}
Los primeros 25 números impares abundantes:
945 suma divisoria adecuada: 975
1575 suma divisoria adecuada: 1649
2205 suma divisoria adecuada: 2241
2835 suma divisoria adecuada: 2973
3465 suma divisoria adecuada: 4023
4095 suma divisoria adecuada: 4641
4725 suma divisoria adecuada: 5195
5355 suma divisoria adecuada: 5877
5775 suma divisoria adecuada: 6129
5985 suma divisoria adecuada: 6495
6435 suma divisoria adecuada: 6669
6615 suma divisoria adecuada: 7065
6825 suma divisoria adecuada: 7063
7245 suma divisoria adecuada: 7731
7425 suma divisoria adecuada: 7455
7875 suma divisoria adecuada: 8349
8085 suma divisoria adecuada: 8331
8415 suma divisoria adecuada: 8433
8505 suma divisoria adecuada: 8967
8925 suma divisoria adecuada: 8931
9135 suma divisoria adecuada: 9585
9555 suma divisoria adecuada: 9597
9765 suma divisoria adecuada: 10203
10395 suma divisoria adecuada: 12645
11025 suma divisoria adecuada: 11946
1000º número impar abundante:
492975 suma divisoria adecuada: 519361
Primer número impar abundante > 1 000 000 000:
1000000575 suma divisoria adecuada: 1083561009
Go
package main import ( "fmt" "strconv" ) func divisors(n int) []int { divs := []int{1} divs2 := []int{} for i := 2; i*i <= n; i++ { if n%i == 0 { j := n / i divs = append(divs, i) if i != j { divs2 = append(divs2, j) } } } for i := len(divs2) - 1; i >= 0; i-- { divs = append(divs, divs2[i]) } return divs } func sum(divs []int) int { tot := 0 for _, div := range divs { tot += div } return tot } func sumStr(divs []int) string { s := "" for _, div := range divs { s += strconv.Itoa(div) + " + " } return s[0 : len(s)-3] } func abundantOdd(searchFrom, countFrom, countTo int, printOne bool) int { count := countFrom n := searchFrom for ; count < countTo; n += 2 { divs := divisors(n) if tot := sum(divs); tot > n { count++ if printOne && count < countTo { continue } s := sumStr(divs) if !printOne { fmt.Printf("%2d. %5d < %s = %d\n", count, n, s, tot) } else { fmt.Printf("%d < %s = %d\n", n, s, tot) } } } return n } func main() { const max = 25 fmt.Println("The first", max, "abundant odd numbers are:") n := abundantOdd(1, 0, 25, false) fmt.Println("\nThe one thousandth abundant odd number is:") abundantOdd(n, 25, 1000, true) fmt.Println("\nThe first abundant odd number above one billion is:") abundantOdd(1e9+1, 0, 1, true) }
{{out}}
The first 25 abundant odd numbers are:
1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Haskell
import Data.List (nub) divisorSum :: Integral a => a -> a divisorSum n = sum . map (\i -> sum $ nub [i, n `quot` i]) . filter ((== 0) . (n `rem`)) $ takeWhile ((<= n) . (^ 2)) [1 ..] oddAbundants :: Integral a => a -> [(a, a)] oddAbundants n = [ (i, divisorSum i) | i <- [n ..], odd i, divisorSum i > i * 2 ] printAbundant :: (Int, Int) -> IO () printAbundant (n, s) = putStrLn $ show n ++ " with " ++ show s ++ " as the sum of all proper divisors." main :: IO () main = do putStrLn "The first 25 odd abundant numbers are:" mapM_ printAbundant . take 25 $ oddAbundants 1 putStrLn "The 1000th odd abundant number is:" printAbundant $ oddAbundants 1 !! 1000 putStrLn "The first odd abundant number above 1000000000 is:" printAbundant . head . oddAbundants $ 10 ^ 9
{{out}}
The first 25 odd abundant numbers are:
945 with 1920 as the sum of all proper divisors.
1575 with 3224 as the sum of all proper divisors.
2205 with 4446 as the sum of all proper divisors.
2835 with 5808 as the sum of all proper divisors.
3465 with 7488 as the sum of all proper divisors.
4095 with 8736 as the sum of all proper divisors.
4725 with 9920 as the sum of all proper divisors.
5355 with 11232 as the sum of all proper divisors.
5775 with 11904 as the sum of all proper divisors.
5985 with 12480 as the sum of all proper divisors.
6435 with 13104 as the sum of all proper divisors.
6615 with 13680 as the sum of all proper divisors.
6825 with 13888 as the sum of all proper divisors.
7245 with 14976 as the sum of all proper divisors.
7425 with 14880 as the sum of all proper divisors.
7875 with 16224 as the sum of all proper divisors.
8085 with 16416 as the sum of all proper divisors.
8415 with 16848 as the sum of all proper divisors.
8505 with 17472 as the sum of all proper divisors.
8925 with 17856 as the sum of all proper divisors.
9135 with 18720 as the sum of all proper divisors.
9555 with 19152 as the sum of all proper divisors.
9765 with 19968 as the sum of all proper divisors.
10395 with 23040 as the sum of all proper divisors.
11025 with 22971 as the sum of all proper divisors.
The 1000th odd abundant number is:
493185 with 1017792 as the sum of all proper divisors.
The first odd abundant number above 1000000000 is:
1000000575 with 2083561584 as the sum of all proper divisors.
Or alternatively (and already a little faster):
import Data.Bool (bool) abundantTuple :: Int -> [(Int, Int)] abundantTuple n = let x = divisorSum n in bool [] [(n, x)] (n < x) divisorSum :: Int -> Int divisorSum n = sum lows + sum (drop (bool 0 1 (iRoot * iRoot == n)) (reverse (quot n <$> tail lows))) where iRoot = floor (sqrt $ fromIntegral n) lows = filter ((== 0) . rem n) [1 .. iRoot] main :: IO () main = do putStrLn "First 25 abundant odd numbers with their divisor sums:" mapM_ print $ take 25 ([1,3 ..] >>= abundantTuple) -- putStrLn "\n1000th odd abundant number with its divisor sum:" print $ ([1,3 ..] >>= abundantTuple) !! 999 -- putStrLn "\nFirst odd abundant number over 10^9, with its divisor sum:" let billion = 10 ^ 9 :: Int print $ head ([1 + billion,3 + billion ..] >>= abundantTuple)
{{Out}}
First 25 abundant odd numbers with their divisor sums:
(945,975)
(1575,1649)
(2205,2241)
(2835,2973)
(3465,4023)
(4095,4641)
(4725,5195)
(5355,5877)
(5775,6129)
(5985,6495)
(6435,6669)
(6615,7065)
(6825,7063)
(7245,7731)
(7425,7455)
(7875,8349)
(8085,8331)
(8415,8433)
(8505,8967)
(8925,8931)
(9135,9585)
(9555,9597)
(9765,10203)
(10395,12645)
(11025,11946)
1000th odd abundant number with its divisor sum:
(492975,519361)
First odd abundant number over 10^9, with its divisor sum:
(1000000575,1083561009)
J
NB. https://www.math.upenn.edu/~deturck/m170/wk3/lecture/sumdiv.html
s=: ([: */ [: ((<:@:(^ >:)/) % <:@:{.) __&q:)&>
assert 6045 -: s 1800
aliquot_sum=: -~ s
abundant=: < aliquot_sum
Filter=: (#~`)(`:6)
A=: abundant Filter 1 2 p. i. 260000 NB. a batch of abundant odd numbers
# A NB. more than 1000, it's enough.
1054
NB. the first odd abundant numbers
(,: aliquot_sum) 26 {. A
945 1575 2205 2835 3465 4095 4725 5355 5775 5985 6435 6615 6825 7245 7425 7875 8085 8415 8505 8925 9135 9555 9765 10395 11025 11655
975 1649 2241 2973 4023 4641 5195 5877 6129 6495 6669 7065 7063 7731 7455 8349 8331 8433 8967 8931 9585 9597 10203 12645 11946 12057
NB. the one thousandth abundant odd number
(,: aliquot_sum) 999 { A
492975
519361
k=: adverb def '1000 * m'
1x k k k
1000000000
abundant Filter (1x k k k) + 1 2x p. i. 10x k
1000000575 1000001475 1000001625 1000001835 1000002465 1000003095 1000003725 1000004355 1000004775 1000004985 1000005435 1000005615 1000005825 1000006245 1000006425 1000006875 1000007505 1000008765 1000009395 1000010025 1000010655 1000011285 1000011705 100...
(,: aliquot_sum) {. abundant Filter (1x k k k) + 1 2x p. i. 10x k
1000000575
1083561009
JavaScript
ES6
Composing reusable functions and generators: {{Trans|Python}}
(() => { 'use strict'; const main = () => { // abundantTuple :: Int -> [(Int, Int)] const abundantTuple = n => { // Either a list containing the tuple of N // and its divisor sum (if n is abundant), // or otherwise an empty list. const x = divisorSum(n); return n < x ? ([ Tuple(n)(x) ]) : []; }; // divisorSum :: Int -> Int const divisorSum = n => { // Sum of the divisors of n. const floatRoot = Math.sqrt(n), intRoot = Math.floor(floatRoot), lows = filter(x => 0 === n % x)( enumFromTo(1)(intRoot) ); return sum(lows.concat(map(quot(n))( intRoot === floatRoot ? ( lows.slice(1, -1) ) : lows.slice(1) ))); }; // TEST --------------------------------------- console.log( 'First 25 abundant odd numbers, with their divisor sums:' ) console.log(unlines(map(showTuple)( take(25)( concatMapGen(abundantTuple)( enumFromThen(1)(3) ) ) ))); console.log( '\n\n1000th abundant odd number, with its divisor sum:' ) console.log(showTuple( take(1)(drop(999)( concatMapGen(abundantTuple)( enumFromThen(1)(3) ) ))[0] )) console.log( '\n\nFirst abundant odd number above 10^9, with divisor sum:' ) const billion = Math.pow(10, 9); console.log(showTuple( take(1)( concatMapGen(abundantTuple)( enumFromThen(1 + billion)(3 + billion) ) )[0] )) }; // GENERAL REUSABLE FUNCTIONS ------------------------- // Tuple (,) :: a -> b -> (a, b) const Tuple = a => b => ({ type: 'Tuple', '0': a, '1': b, length: 2 }); // concatMapGen :: (a -> [b]) -> Gen [a] -> Gen [b] const concatMapGen = f => function*(xs) { let x = xs.next(), v = undefined; while (!x.done) { v = f(x.value); if (0 < v.length) { yield v[0]; } x = xs.next(); } }; // drop :: Int -> [a] -> [a] // drop :: Int -> Generator [a] -> Generator [a] // drop :: Int -> String -> String const drop = n => xs => Infinity > length(xs) ? ( xs.slice(n) ) : (take(n)(xs), xs); // dropAround :: (a -> Bool) -> [a] -> [a] // dropAround :: (Char -> Bool) -> String -> String const dropAround = p => xs => dropWhile(p)( dropWhileEnd(p)(xs) ); // dropWhile :: (a -> Bool) -> [a] -> [a] // dropWhile :: (Char -> Bool) -> String -> String const dropWhile = p => xs => { const lng = xs.length; return 0 < lng ? xs.slice( until(i => i === lng || !p(xs[i]))( i => 1 + i )(0) ) : []; }; // dropWhileEnd :: (a -> Bool) -> [a] -> [a] // dropWhileEnd :: (Char -> Bool) -> String -> String const dropWhileEnd = p => xs => { let i = xs.length; while (i-- && p(xs[i])) {} return xs.slice(0, i + 1); }; // enumFromThen :: Int -> Int -> Gen [Int] const enumFromThen = x => // A non-finite stream of integers, // starting with x and y, and continuing // with the same interval. function*(y) { const d = y - x; let v = y + d; yield x; yield y; while (true) { yield v; v = d + v; } }; // enumFromTo :: Int -> Int -> [Int] const enumFromTo = m => n => Array.from({ length: 1 + n - m }, (_, i) => m + i); // filter :: (a -> Bool) -> [a] -> [a] const filter = f => xs => xs.filter(f); // Returns Infinity over objects without finite length. // This enables zip and zipWith to choose the shorter // argument when one is non-finite, like cycle, repeat etc // length :: [a] -> Int const length = xs => (Array.isArray(xs) || 'string' === typeof xs) ? ( xs.length ) : Infinity; // map :: (a -> b) -> [a] -> [b] const map = f => xs => (Array.isArray(xs) ? ( xs ) : xs.split('')).map(f); // quot :: Int -> Int -> Int const quot = n => m => Math.floor(n / m); // show :: a -> String const show = JSON.stringify; // showTuple :: Tuple -> String const showTuple = tpl => '(' + enumFromTo(0)(tpl.length - 1) .map(x => unQuoted(show(tpl[x]))) .join(',') + ')'; // sum :: [Num] -> Num const sum = xs => xs.reduce((a, x) => a + x, 0); // take :: Int -> [a] -> [a] // take :: Int -> String -> String const take = n => xs => 'GeneratorFunction' !== xs.constructor.constructor.name ? ( xs.slice(0, n) ) : [].concat.apply([], Array.from({ length: n }, () => { const x = xs.next(); return x.done ? [] : [x.value]; })); // unlines :: [String] -> String const unlines = xs => xs.join('\n'); // until :: (a -> Bool) -> (a -> a) -> a -> a const until = p => f => x => { let v = x; while (!p(v)) v = f(v); return v; }; // unQuoted :: String -> String const unQuoted = s => dropAround(x => 34 === x.codePointAt(0))( s ); // MAIN --- return main(); })();
{{Out}}
First 25 abundant odd numbers, with their divisor sums:
(945,975)
(1575,1649)
(2205,2241)
(2835,2973)
(3465,4023)
(4095,4641)
(4725,5195)
(5355,5877)
(5775,6129)
(5985,6495)
(6435,6669)
(6615,7065)
(6825,7063)
(7245,7731)
(7425,7455)
(7875,8349)
(8085,8331)
(8415,8433)
(8505,8967)
(8925,8931)
(9135,9585)
(9555,9597)
(9765,10203)
(10395,12645)
(11025,11946)
1000th abundant odd number, with its divisor sum:
(492975,519361)
First abundant odd number above 10^9, with divisor sum:
(1000000575,1083561009)
Julia
using Primes function propfact(n) f = [one(n)] for (p, x) in factor(n) f = reduce(vcat, [f*p^i for i in 1:x], init=f) end pop!(f) f end isabundant(n) = sum(propfact(n)) > n prettyprintfactors(n) = (a = propfact(n); println("$n has proper divisors $a, these sum to $(sum(a)).")) function oddabundantsfrom(startingint, needed, nprint=0) n = isodd(startingint) ? startingint : startingint + 1 count = 0 while count < needed if isabundant(n) if nprint == 0 prettyprintfactors(n) elseif nprint == count + 1 prettyprintfactors(n) break end count += 1 end n += 2 end end println("First 25 abundant odd numbers:") oddabundantsfrom(2, 25) println("The thousandth abundant odd number:") oddabundantsfrom(2, 1001, 1000) println("The first abundant odd number greater than one billion:") oddabundantsfrom(1000000000, 1)
{{out}}
First 25 abundant odd numbers:
945 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315], these sum to 975.
1575 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525], these sum to 1649.
2205 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 49, 147, 441, 245, 735], these sum to 2241.
2835 has proper divisors [1, 3, 9, 27, 81, 5, 15, 45, 135, 405, 7, 21, 63, 189, 567, 35, 105, 315, 945], these sum to 2973.
3465 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 11, 33, 99, 55, 165, 495, 77, 231, 693, 385, 1155], these sum to 4023.
4095 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 13, 39, 117, 65, 195, 585, 91, 273, 819, 455, 1365], these sum to 4641.
4725 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675, 7, 21, 63, 189, 35, 105, 315, 945, 175, 525, 1575], these sum to 5195.
5355 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 17, 51, 153, 85, 255, 765, 119, 357, 1071, 595, 1785], these sum to 5877.
5775 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 11, 33, 55, 165, 275, 825, 77, 231, 385, 1155, 1925], these sum to 6129.
5985 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 19, 57, 171, 95, 285, 855, 133, 399, 1197, 665, 1995], these sum to 6495.
6435 has proper divisors [1, 3, 9, 5, 15, 45, 11, 33, 99, 55, 165, 495, 13, 39, 117, 65, 195, 585, 143, 429, 1287, 715, 2145], these sum to 6669.
6615 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315, 945, 49, 147, 441, 1323, 245, 735, 2205], these sum to 7065.
6825 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 13, 39, 65, 195, 325, 975, 91, 273, 455, 1365, 2275], these sum to 7063.
7245 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 23, 69, 207, 115, 345, 1035, 161, 483, 1449, 805, 2415], these sum to 7731.
7425 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675, 11, 33, 99, 297, 55, 165, 495, 1485, 275, 825, 2475], these sum to 7455.
7875 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 125, 375, 1125, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 875, 2625], these sum to 8349.
8085 has proper divisors [1, 3, 5, 15, 7, 21, 35, 105, 49, 147, 245, 735, 11, 33, 55, 165, 77, 231, 385, 1155, 539, 1617, 2695], these sum to 8331.
8415 has proper divisors [1, 3, 9, 5, 15, 45, 11, 33, 99, 55, 165, 495, 17, 51, 153, 85, 255, 765, 187, 561, 1683, 935, 2805], these sum to 8433.
8505 has proper divisors [1, 3, 9, 27, 81, 243, 5, 15, 45, 135, 405, 1215, 7, 21, 63, 189, 567, 1701, 35, 105, 315, 945, 2835], these sum to 8967.
8925 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 17, 51, 85, 255, 425, 1275, 119, 357, 595, 1785, 2975], these sum to 8931.
9135 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 29, 87, 261, 145, 435, 1305, 203, 609, 1827, 1015, 3045], these sum to 9585.
9555 has proper divisors [1, 3, 5, 15, 7, 21, 35, 105, 49, 147, 245, 735, 13, 39, 65, 195, 91, 273, 455, 1365, 637, 1911, 3185], these sum to 9597.
9765 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 31, 93, 279, 155, 465, 1395, 217, 651, 1953, 1085, 3255], these sum to 10203.
10395 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315, 945, 11, 33, 99, 297, 55, 165, 495, 1485, 77, 231, 693, 2079, 385, 1155, 3465], these sum to 12645.
11025 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 49, 147, 441, 245, 735, 2205, 1225, 3675], these sum to 11946.
The thousandth abundant odd number:
492975 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 313, 939, 2817, 1565, 4695, 14085, 7825, 23475, 70425, 2191, 6573, 19719, 10955, 32865, 98595, 54775, 164325], these sum to 519361.
The first abundant odd number greater than one billion:
1000000575 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 49, 147, 441, 245, 735, 2205, 1225, 3675, 11025, 90703, 272109, 816327, 453515, 1360545, 4081635, 2267575, 6802725, 20408175, 634921, 1904763, 5714289, 3174605, 9523815, 28571445, 15873025, 47619075, 142857225, 4444447, 13333341, 40000023, 22222235, 66666705, 200000115, 111111175, 333333525], these sum to 1083561009.
Kotlin
{{trans|D}}
fun divisors(n: Int): List<Int> { val divs = mutableListOf(1) val divs2 = mutableListOf<Int>() var i = 2 while (i * i <= n) { if (n % i == 0) { val j = n / i divs.add(i) if (i != j) { divs2.add(j) } } i++ } divs.addAll(divs2.reversed()) return divs } fun abundantOdd(searchFrom: Int, countFrom: Int, countTo: Int, printOne: Boolean): Int { var count = countFrom var n = searchFrom while (count < countTo) { val divs = divisors(n) val tot = divs.sum() if (tot > n) { count++ if (!printOne || count >= countTo) { val s = divs.joinToString(" + ") if (printOne) { println("$n < $s = $tot") } else { println("%2d. %5d < %s = %d".format(count, n, s, tot)) } } } n += 2 } return n } fun main() { val max = 25 println("The first $max abundant odd numbers are:") val n = abundantOdd(1, 0, 25, false) println("\nThe one thousandth abundant odd number is:") abundantOdd(n, 25, 1000, true) println("\nThe first abundant odd number above one billion is:") abundantOdd((1e9 + 1).toInt(), 0, 1, true) }
{{out}}
The first 25 abundant odd numbers are:
1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Lua
-- Return the sum of the proper divisors of x function sumDivs (x) local sum, sqr = 1, math.sqrt(x) for d = 2, sqr do if x % d == 0 then sum = sum + d if d ~= sqr then sum = sum + (x/d) end end end return sum end -- Return a table of odd abundant numbers function oddAbundants (mode, limit) local n, count, divlist, divsum = 1, 0, {} repeat n = n + 2 divsum = sumDivs(n) if divsum > n then table.insert(divlist, {n, divsum}) count = count + 1 if mode == "Above" and n > limit then return divlist[#divlist] end end until count == limit if mode == "First" then return divlist end if mode == "Nth" then return divlist[#divlist] end end -- Write a result to stdout function showResult (msg, t) print(msg .. ": the proper divisors of " .. t[1] .. " sum to " .. t[2]) end -- Main procedure for k, v in pairs(oddAbundants("First", 25)) do showResult(k, v) end showResult("1000", oddAbundants("Nth", 1000)) showResult("Above 1e6", oddAbundants("Above", 1e6))
{{out}}
1: the proper divisors of 945 sum to 975
2: the proper divisors of 1575 sum to 1649
3: the proper divisors of 2205 sum to 2241
4: the proper divisors of 2835 sum to 2973
5: the proper divisors of 3465 sum to 4023
6: the proper divisors of 4095 sum to 4641
7: the proper divisors of 4725 sum to 5195
8: the proper divisors of 5355 sum to 5877
9: the proper divisors of 5775 sum to 6129
10: the proper divisors of 5985 sum to 6495
11: the proper divisors of 6435 sum to 6669
12: the proper divisors of 6615 sum to 7065
13: the proper divisors of 6825 sum to 7063
14: the proper divisors of 7245 sum to 7731
15: the proper divisors of 7425 sum to 7455
16: the proper divisors of 7875 sum to 8349
17: the proper divisors of 8085 sum to 8331
18: the proper divisors of 8415 sum to 8433
19: the proper divisors of 8505 sum to 8967
20: the proper divisors of 8925 sum to 8931
21: the proper divisors of 9135 sum to 9585
22: the proper divisors of 9555 sum to 9597
23: the proper divisors of 9765 sum to 10203
24: the proper divisors of 10395 sum to 12645
25: the proper divisors of 11025 sum to 11946
1000: the proper divisors of 492975 sum to 519361
Above 1e6: the proper divisors of 1000125 sum to 1076547
Perl
{{trans|Perl 6}} {{libheader|ntheory}}
use strict; use warnings; use feature 'say'; use ntheory qw/divisor_sum divisors/; sub odd_abundants { my($start,$count) = @_; my $n = int(( $start + 2 ) / 3); $n += 1 if 0 == $n % 2; $n *= 3; my @out; while (@out < $count) { $n += 6; next unless (my $ds = divisor_sum($n)) > 2*$n; my @d = divisors($n); push @out, sprintf "%6d: divisor sum: %s = %d", $n, join(' + ', @d[0..@d-2]), $ds-$n; } @out; } say 'First 25 abundant odd numbers:'; say for odd_abundants(1, 25); say "\nOne thousandth abundant odd number:\n", (odd_abundants(1, 1000))[999]; say "\nFirst abundant odd number above one billion:\n", odd_abundants(999_999_999, 1);
{{out}}
First 25 abundant odd numbers:
945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
One thousandth abundant odd number:
492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
First abundant odd number above one billion:
1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
```
## Perl 6
{{works with|Rakudo|2019.03}}
```perl6
sub odd-abundant (\x) {
my @l = x.is-prime ?? 1 !! flat
1, (3 .. x.sqrt.floor).map: -> \d {
next unless d +& 1;
my \y = x div d;
next if y * d !== x;
d !== y ?? (d, y) !! d
};
@l.sum > x ?? @l.sort !! Empty;
}
sub odd-abundants (Int :$start-at is copy) {
$start-at = ( $start-at + 2 ) div 3;
$start-at += $start-at %% 2;
$start-at *= 3;
($start-at, *+6 ... *).hyper.map: {
next unless my $oa = .&odd-abundant;
sprintf "%6d: divisor sum: {$oa.join: ' + '} = {$oa.sum}", $_
}
}
put 'First 25 abundant odd numbers:';
.put for odd-abundants( :start-at(1) )[^25];
put "\nOne thousandth abundant odd number:\n" ~ odd-abundants( :start-at(1) )[999] ~
"\n\nFirst abundant odd number above one billion:\n" ~ odd-abundants( :start-at(1_000_000_000) ).head;
```
{{out}}
```txt
First 25 abundant odd numbers:
945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
One thousandth abundant odd number:
492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
First abundant odd number above one billion:
1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
```
## Phix
```Phix
function abundantOdd(integer n, done, lim, bool printAll)
while donen then
done += 1
if printAll or done=lim then
string ln = iff(printAll?sprintf("%2d. ",done):"")
printf(1,"%s%,6d (proper sum:%,d)\n",{ln,n,tot})
end if
end if
n += 2
end while
printf(1,"\n")
return n
end function
printf(1,"The first 25 abundant odd numbers are:\n")
integer n = abundantOdd(1, 0, 25, true)
printf(1,"The one thousandth abundant odd number is:")
{} = abundantOdd(n, 25, 1000, false)
printf(1,"The first abundant odd number above one billion is:")
{} = abundantOdd(1e9+1, 0, 1, false)
```
{{out}}
```txt
The first 25 abundant odd numbers are:
1. 945 (proper sum:975)
2. 1,575 (proper sum:1,649)
3. 2,205 (proper sum:2,241)
4. 2,835 (proper sum:2,973)
5. 3,465 (proper sum:4,023)
6. 4,095 (proper sum:4,641)
7. 4,725 (proper sum:5,195)
8. 5,355 (proper sum:5,877)
9. 5,775 (proper sum:6,129)
10. 5,985 (proper sum:6,495)
11. 6,435 (proper sum:6,669)
12. 6,615 (proper sum:7,065)
13. 6,825 (proper sum:7,063)
14. 7,245 (proper sum:7,731)
15. 7,425 (proper sum:7,455)
16. 7,875 (proper sum:8,349)
17. 8,085 (proper sum:8,331)
18. 8,415 (proper sum:8,433)
19. 8,505 (proper sum:8,967)
20. 8,925 (proper sum:8,931)
21. 9,135 (proper sum:9,585)
22. 9,555 (proper sum:9,597)
23. 9,765 (proper sum:10,203)
24. 10,395 (proper sum:12,645)
25. 11,025 (proper sum:11,946)
The one thousandth abundant odd number is:492,975 (proper sum:519,361)
The first abundant odd number above one billion is:1,000,000,575 (proper sum:1,083,561,009)
```
## PicoLisp
```PicoLisp
(de accud (Var Key)
(if (assoc Key (val Var))
(con @ (inc (cdr @)))
(push Var (cons Key 1)) )
Key )
(de **sum (L)
(let S 1
(for I (cdr L)
(inc 'S (** (car L) I)) )
S ) )
(de factor-sum (N)
(if (=1 N)
0
(let
(R NIL
D 2
L (1 2 2 . (4 2 4 2 4 6 2 6 .))
M (sqrt N)
N1 N
S 1 )
(while (>= M D)
(if (=0 (% N1 D))
(setq M
(sqrt (setq N1 (/ N1 (accud 'R D)))) )
(inc 'D (pop 'L)) ) )
(accud 'R N1)
(for I R
(setq S (* S (**sum I))) )
(- S N) ) ) )
(de factor-list NIL
(let (N 1 C 0)
(make
(loop
(when (> (setq @@ (factor-sum N)) N)
(link (cons N @@))
(inc 'C) )
(inc 'N 2)
(T (= C 1000)) ) ) ) )
(let L (factor-list)
(for N 25
(println N (++ L)) )
(println 1000 (last L))
(println
'****
1000000575
(factor-sum 1000000575) ) )
```
{{out}}
```txt
1 (945 . 975)
2 (1575 . 1649)
3 (2205 . 2241)
4 (2835 . 2973)
5 (3465 . 4023)
6 (4095 . 4641)
7 (4725 . 5195)
8 (5355 . 5877)
9 (5775 . 6129)
10 (5985 . 6495)
11 (6435 . 6669)
12 (6615 . 7065)
13 (6825 . 7063)
14 (7245 . 7731)
15 (7425 . 7455)
16 (7875 . 8349)
17 (8085 . 8331)
18 (8415 . 8433)
19 (8505 . 8967)
20 (8925 . 8931)
21 (9135 . 9585)
22 (9555 . 9597)
23 (9765 . 10203)
24 (10395 . 12645)
25 (11025 . 11946)
1000 (492975 . 519361)
**** 1000000575 1083561009
```
## PureBasic
{{trans|C}}
```PureBasic
NewList l_sum.i()
Procedure.i sum_proper_divisors(n.i)
Define.i sum, i=3, j
Shared l_sum()
AddElement(l_sum())
l_sum()=1
While ij
sum+j
AddElement(l_sum())
l_sum()=j
EndIf
EndIf
i+2
Wend
ProcedureReturn sum+1
EndProcedure
If OpenConsole("Abundant_odd_numbers")
Define.i n, c, s
n=1
c=0
While c<25
ClearList(l_sum())
s=sum_proper_divisors(n)
If n "+RSet(Str(s),6))
ForEach l_sum()
If ListIndex(l_sum())=0
Print(" = ")
Else
Print("+")
EndIf
Print(Str(l_sum()))
Next
PrintN("")
EndIf
n+2
Wend
n-2
While c<1000
s=sum_proper_divisors(n+2)
c+Bool(n oddNumber :
aCount += 1
print("{0:5} proper divisor sum: {1}". format(oddNumber ,dSum ))
oddNumber += 2
while aCount < 1000:
dSum = divisorSum(oddNumber )
if dSum > oddNumber :
aCount += 1
oddNumber += 2
print ("\n1000th abundant odd number:")
print (" ",(oddNumber - 2)," proper divisor sum: ",dSum)
oddNumber = 1000000001
found = False
while not found :
dSum = divisorSum(oddNumber )
if dSum > oddNumber :
found = True
print ("\nFirst abundant odd number > 1 000 000 000:")
print (" ",oddNumber," proper divisor sum: ",dSum)
oddNumber += 2
```
{{out}}
```txt
The first 25 abundant odd numbers:
945 proper divisor sum: 975
1575 proper divisor sum: 1649
2205 proper divisor sum: 2241
2835 proper divisor sum: 2973
3465 proper divisor sum: 4023
4095 proper divisor sum: 4513
4725 proper divisor sum: 5195
5355 proper divisor sum: 5877
5775 proper divisor sum: 5977
5985 proper divisor sum: 6495
6435 proper divisor sum: 6669
6615 proper divisor sum: 7065
6825 proper divisor sum: 7063
7245 proper divisor sum: 7731
7425 proper divisor sum: 7455
7875 proper divisor sum: 8349
8085 proper divisor sum: 8331
8415 proper divisor sum: 8433
8505 proper divisor sum: 8967
8925 proper divisor sum: 8931
9135 proper divisor sum: 9585
9555 proper divisor sum: 9597
9765 proper divisor sum: 10203
10395 proper divisor sum: 12645
11025 proper divisor sum: 11946
1000th abundant odd number:
492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
1000000575 proper divisor sum: 1083561009
```
### Functional
```python
'''Odd abundant numbers'''
from math import sqrt
from itertools import chain, count, islice
# abundantTuple :: Int -> [(Int, Int)]
def abundantTuple(n):
'''A list containing the tuple of N and its divisor
sum, if n is abundant, or an empty list.
'''
x = divisorSum(n)
return [(n, x)] if n < x else []
# divisorSum :: Int -> Int
def divisorSum(n):
'''Sum of the divisors of n.'''
floatRoot = sqrt(n)
intRoot = int(floatRoot)
blnSquare = intRoot == floatRoot
lows = [x for x in range(1, 1 + intRoot) if 0 == n % x]
return sum(lows + [
n // x for x in (
lows[1:-1] if blnSquare else lows[1:]
)
])
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''Subsets of abundant odd numbers.'''
# First 25.
print('First 25 abundant odd numbers with their divisor sums:')
for x in take(25)(
concatMap(abundantTuple)(
enumFromThen(1)(3)
)
):
print(x)
# The 1000th.
print('\n1000th odd abundant number with its divisor sum:')
print(
take(1000)(
concatMap(abundantTuple)(
enumFromThen(1)(3)
)
)[-1]
)
# First over 10^9.
print('\nFirst odd abundant number over 10^9, with its divisor sum:')
billion = (10 ** 9)
print(
take(1)(
concatMap(abundantTuple)(
enumFromThen(1 + billion)(3 + billion)
)
)[0]
)
# GENERAL FUNCTIONS ---------------------------------------
# enumFromThen :: Int -> Int -> [Int]
def enumFromThen(m):
'''A non-finite stream of integers
starting at m, and continuing
at the interval between m and n.
'''
return lambda n: count(m, n - m)
# concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f):
'''A concatenated list over which a function f
has been mapped.
The list monad can be derived by using an (a -> [b])
function which wraps its output in a list (using an
empty list to represent computational failure).
'''
return lambda xs: (
chain.from_iterable(map(f, xs))
)
# take :: Int -> [a] -> [a]
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.
'''
return lambda xs: (
list(islice(xs, n))
)
if __name__ == '__main__':
main()
```
{{Out}}
```txt
First 25 abundant odd numbers with their divisor sums:
(945, 975)
(1575, 1649)
(2205, 2241)
(2835, 2973)
(3465, 4023)
(4095, 4641)
(4725, 5195)
(5355, 5877)
(5775, 6129)
(5985, 6495)
(6435, 6669)
(6615, 7065)
(6825, 7063)
(7245, 7731)
(7425, 7455)
(7875, 8349)
(8085, 8331)
(8415, 8433)
(8505, 8967)
(8925, 8931)
(9135, 9585)
(9555, 9597)
(9765, 10203)
(10395, 12645)
(11025, 11946)
1000th odd abundant number with its divisor sum:
(492975, 519361)
First odd abundant number over 10^9, with its divisor sum:
(1000000575, 1083561009)
```
## Racket
```racket
#lang racket
(require math/number-theory
racket/generator)
(define (make-generator start)
(in-generator
(for ([n (in-naturals start)] #:when (odd? n))
(define divisor-sum (- (apply + (divisors n)) n))
(when (> divisor-sum n) (yield (list n divisor-sum))))))
(for/list ([i (in-range 25)] [x (make-generator 0)]) x) ; Task 1
(for/last ([i (in-range 1000)] [x (make-generator 0)]) x) ; Task 2
(for/first ([x (make-generator (add1 (inexact->exact 1e9)))]) x) ; Task 3
```
{{out}}
```txt
'((945 975)
(1575 1649)
(2205 2241)
(2835 2973)
(3465 4023)
(4095 4641)
(4725 5195)
(5355 5877)
(5775 6129)
(5985 6495)
(6435 6669)
(6615 7065)
(6825 7063)
(7245 7731)
(7425 7455)
(7875 8349)
(8085 8331)
(8415 8433)
(8505 8967)
(8925 8931)
(9135 9585)
(9555 9597)
(9765 10203)
(10395 12645)
(11025 11946))
'(492975 519361)
'(1000000575 1083561009)
```
## REXX
A wee bit of coding was added to add commas to numbers (because of the larger numbers) as well as alignment of the output.
The '''sigO''' is a specialized version of '''sigma''' optimized just for odd numbers.
```rexx
/*REXX pgm displays abundant odd numbers: 1st 25, one─thousandth, first > 1 billion. */
parse arg Nlow Nuno Novr . /*obtain optional arguments from the CL*/
if Nlow=='' | Nlow=="," then Nlow= 25 /*Not specified? Then use the default.*/
if Nuno=='' | Nuno=="," then Nuno= 1000 /* " " " " " " */
if Novr=='' | Novr=="," then Novr= 1000000000 /* " " " " " " */
numeric digits max(9, length(Novr) ) /*ensure enough decimal digits for // */
@= 'odd abundant number' /*variable for annotating the output. */
# = 0 /*count of odd abundant numbers so far.*/
do j=3 by 2 until #>=Nlow; $= sigO(j) /*get the sigma for an odd integer. */
if $<=j then iterate /*sigma ≤ J ? Then ignore it. */
#= # + 1 /*bump the counter for abundant odd #'s*/
say rt(th(#)) @ 'is:'rt(commas(j), 8) rt("sigma=") rt(commas($), 9)
end /*j*/
say
# = 0 /*count of odd abundant numbers so far.*/
do j=3 by 2; $= sigO(j) /*get the sigma for an odd integer. */
if $<=j then iterate /*sigma ≤ J ? Then ignore it. */
#= # + 1 /*bump the counter for abundant odd #'s*/
if # n
check = 1
index = index + 1
if index < limit + 1
showArray(n,nArray,sum,index)
ok
if index = 100
see "One thousandth abundant odd number:" + nl
showArray2(n,nArray,sum,index)
ok
if index = 100000000
see "First abundant odd number above one billion:" + nl
showArray2(n,nArray,sum,index)
ok
ok
func showArray(n,nArray,sum,index)
see "" + index + ". " + string(n) + ": divisor sum: "
for m = 1 to len(nArray)
if m < len(nArray)
see string(nArray[m]) + " + "
else
see string(nArray[m]) + " = " + string(sum) + nl + nl
ok
next
func showArray2(n,nArray,sum,index)
see "" + index + ". " + string(n) + ": divisor sum: " +
see string(nArray[m]) + " = " + string(sum) + nl + nl
```
```txt
working...
wait for done...
1. 945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
One thousandth abundant odd number:
1000. 492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
First abundant odd number above one billion:
100000000. 1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
done...
```
## Ruby
proper_divisors method taken from http://rosettacode.org/wiki/Proper_divisors#Ruby
```ruby
require "prime"
class Integer
def proper_divisors
return [] if self == 1
primes = prime_division.flat_map{|prime, freq| [prime] * freq}
(1...primes.size).each_with_object([1]) do |n, res|
primes.combination(n).map{|combi| res << combi.inject(:*)}
end.flatten.uniq
end
end
def generator_odd_abundants(from=1)
from += 1 if from.even?
Enumerator.new do |y|
from.step(nil, 2) do |n|
sum = n.proper_divisors.sum
y << [n, sum] if sum > n
end
end
end
generator_odd_abundants.take(25).each{|n, sum| puts "#{n} with sum #{sum}" }
puts "\n%d with sum %#d" % generator_odd_abundants.take(1000).last
puts "\n%d with sum %#d" % generator_odd_abundants(1_000_000_000).next
```
## Rust
{{trans|Go}}
```rust
fn divisors(n: u64) -> Vec {
let mut divs = vec![1];
let mut divs2 = Vec::new();
for i in (2..).take_while(|x| x * x <= n).filter(|x| n % x == 0) {
divs.push(i);
let j = n / i;
if i != j {
divs2.push(j);
}
}
divs.extend(divs2.iter().rev());
divs
}
fn sum_string(v: Vec) -> String {
v[1..]
.iter()
.fold(format!("{}", v[0]), |s, i| format!("{} + {}", s, i))
}
fn abundant_odd(search_from: u64, count_from: u64, count_to: u64, print_one: bool) -> u64 {
let mut count = count_from;
for n in (search_from..).step_by(2) {
let divs = divisors(n);
let total: u64 = divs.iter().sum();
if total > n {
count += 1;
let s = sum_string(divs);
if !print_one {
println!("{}. {} < {} = {}", count, n, s, total);
} else if count == count_to {
println!("{} < {} = {}", n, s, total);
}
}
if count == count_to {
break;
}
}
count_to
}
fn main() {
let max = 25;
println!("The first {} abundant odd numbers are:", max);
let n = abundant_odd(1, 0, max, false);
println!("The one thousandth abundant odd number is:");
abundant_odd(n, 25, 1000, true);
println!("The first abundant odd number above one billion is:");
abundant_odd(1e9 as u64 + 1, 0, 1, true);
}
```
{{out}}
```txt
The first 25 abundant odd numbers are:
1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
The one thousandth abundant odd number is:
479115 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 27 + 35 + 39 + 45 + 63 + 65 + 81 + 91 + 105 + 117 + 135 + 169 + 189 + 195 + 273 + 315 + 351 + 405 + 455 + 507 + 567 + 585 + 819 + 845 + 945 + 1053 + 1183 + 1365 + 1521 + 1755 + 2457 + 2535 + 2835 + 3549 + 4095 + 4563 + 5265 + 5915 + 7371 + 7605 + 10647 + 12285 + 13689 + 17745 + 22815 + 31941 + 36855 + 53235 + 68445 + 95823 + 159705 = 583749
The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
```
## Scala
{{trans|D}}
```scala
import scala.collection.mutable.ListBuffer
object Abundant {
def divisors(n: Int): ListBuffer[Int] = {
val divs = new ListBuffer[Int]
divs.append(1)
val divs2 = new ListBuffer[Int]
var i = 2
while (i * i <= n) {
if (n % i == 0) {
val j = n / i
divs.append(i)
if (i != j) {
divs2.append(j)
}
}
i += 1
}
divs.appendAll(divs2.reverse)
divs
}
def abundantOdd(searchFrom: Int, countFrom: Int, countTo: Int, printOne: Boolean): Int = {
var count = countFrom
var n = searchFrom
while (count < countTo) {
val divs = divisors(n)
val tot = divs.sum
if (tot > n) {
count += 1
if (!printOne || !(count < countTo)) {
val s = divs.map(a => a.toString).mkString(" + ")
if (printOne) {
printf("%d < %s = %d\n", n, s, tot)
} else {
printf("%2d. %5d < %s = %d\n", count, n, s, tot)
}
}
}
n += 2
}
n
}
def main(args: Array[String]): Unit = {
val max = 25
printf("The first %d abundant odd numbers are:\n", max)
val n = abundantOdd(1, 0, max, printOne = false)
printf("\nThe one thousandth abundant odd number is:\n")
abundantOdd(n, 25, 1000, printOne = true)
printf("\nThe first abundant odd number above one billion is:\n")
abundantOdd((1e9 + 1).intValue(), 0, 1, printOne = true)
}
}
```
{{out}}
```txt
The first 25 abundant odd numbers are:
1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975
2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649
3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241
4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973
5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023
6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641
7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195
8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877
9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129
10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495
11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669
12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065
13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063
14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731
15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455
16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349
17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331
18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433
19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967
20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931
21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585
22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597
23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203
24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645
25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946
The one thousandth abundant odd number is:
492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361
The first abundant odd number above one billion is:
1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
```
## Sidef
```ruby
func is_abundant(n) {
n.sigma > 2*n
}
func odd_abundants (from = 1) {
from = (from + 2)//3
from += (from%2 - 1)
3*from .. Inf `by` 6 -> lazy.grep(is_abundant)
}
say " Index | Number | proper divisor sum"
const sep = "-------+-------------+-------------------\n"
const fstr = "%6s | %11s | %11s\n"
print sep
odd_abundants().first(25).each_kv {|k,n|
printf(fstr, k+1, n, n.sigma-n)
}
with (odd_abundants().nth(1000)) {|n|
printf(sep + fstr, 1000, n, n.sigma-n)
}
with(odd_abundants(1e9).first) {|n|
printf(sep + fstr, '***', n, n.sigma-n)
}
```
{{out}}
```txt
Index | Number | proper divisor sum
-------+-------------+-------------------
1 | 945 | 975
2 | 1575 | 1649
3 | 2205 | 2241
4 | 2835 | 2973
5 | 3465 | 4023
6 | 4095 | 4641
7 | 4725 | 5195
8 | 5355 | 5877
9 | 5775 | 6129
10 | 5985 | 6495
11 | 6435 | 6669
12 | 6615 | 7065
13 | 6825 | 7063
14 | 7245 | 7731
15 | 7425 | 7455
16 | 7875 | 8349
17 | 8085 | 8331
18 | 8415 | 8433
19 | 8505 | 8967
20 | 8925 | 8931
21 | 9135 | 9585
22 | 9555 | 9597
23 | 9765 | 10203
24 | 10395 | 12645
25 | 11025 | 11946
-------+-------------+-------------------
1000 | 492975 | 519361
-------+-------------+-------------------
*** | 1000000575 | 1083561009
```
## Visual Basic .NET
{{Trans|ALGOL 68}}
```vbnet
Module AbundantOddNumbers
' find some abundant odd numbers - numbers where the sum of the proper
' divisors is bigger than the number
' itself
' returns the sum of the proper divisors of n
Private Function divisorSum(n As Integer) As Integer
Dim sum As Integer = 1
For d As Integer = 2 To Math.Round(Math.Sqrt(n))
If n Mod d = 0 Then
sum += d
Dim otherD As Integer = n \ d
IF otherD <> d Then
sum += otherD
End If
End If
Next d
Return sum
End Function
' find numbers required by the task
Public Sub Main(args() As String)
' first 25 odd abundant numbers
Dim oddNumber As Integer = 1
Dim aCount As Integer = 0
Dim dSum As Integer = 0
Console.Out.WriteLine("The first 25 abundant odd numbers:")
Do While aCount < 25
dSum = divisorSum(oddNumber)
If dSum > oddNumber Then
aCount += 1
Console.Out.WriteLine(oddNumber.ToString.PadLeft(6) & " proper divisor sum: " & dSum)
End If
oddNumber += 2
Loop
' 1000th odd abundant number
Do While aCount < 1000
dSum = divisorSum(oddNumber)
If dSum > oddNumber Then
aCount += 1
End If
oddNumber += 2
Loop
Console.Out.WriteLine("1000th abundant odd number:")
Console.Out.WriteLine(" " & (oddNumber - 2) & " proper divisor sum: " & dSum)
' first odd abundant number > one billion
oddNumber = 1000000001
Dim found As Boolean = False
Do While Not found
dSum = divisorSum(oddNumber)
If dSum > oddNumber Then
found = True
Console.Out.WriteLine("First abundant odd number > 1 000 000 000:")
Console.Out.WriteLine(" " & oddNumber & " proper divisor sum: " & dSum)
End If
oddNumber += 2
Loop
End Sub
End Module
```
{{out}}
```txt
The first 25 abundant odd numbers:
945 proper divisor sum: 975
1575 proper divisor sum: 1649
2205 proper divisor sum: 2241
2835 proper divisor sum: 2973
3465 proper divisor sum: 4023
4095 proper divisor sum: 4641
4725 proper divisor sum: 5195
5355 proper divisor sum: 5877
5775 proper divisor sum: 6129
5985 proper divisor sum: 6495
6435 proper divisor sum: 6669
6615 proper divisor sum: 7065
6825 proper divisor sum: 7063
7245 proper divisor sum: 7731
7425 proper divisor sum: 7455
7875 proper divisor sum: 8349
8085 proper divisor sum: 8331
8415 proper divisor sum: 8433
8505 proper divisor sum: 8967
8925 proper divisor sum: 8931
9135 proper divisor sum: 9585
9555 proper divisor sum: 9597
9765 proper divisor sum: 10203
10395 proper divisor sum: 12645
11025 proper divisor sum: 11946
1000th abundant odd number:
492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
1000000575 proper divisor sum: 1083561009
```
## zkl
```zkl
fcn oddAbundants(startAt=3){ //--> iterator
Walker.zero().tweak(fcn(rn){
n:=rn.value;
while(True){
sum:=0;
foreach d in ([3.. n.toFloat().sqrt().toInt(), 2]){
if( (y:=n/d) *d != n) continue;
sum += ((y==d) and y or y+d)
}
if(sum>n){ rn.set(n+2); return(n) }
n+=2;
}
}.fp(Ref(startAt.isOdd and startAt or startAt+1)))
}
```
```zkl
fcn oddDivisors(n){ // -->sorted List
[3.. n.toFloat().sqrt().toInt(), 2].pump(List(1),'wrap(d){
if( (y:=n/d) *d != n) return(Void.Skip);
if (y==d) y else T(y,d)
}).flatten().sort()
}
fcn printOAs(oas){ // List | int
foreach n in (vm.arglist.flatten()){
ds:=oddDivisors(n);
println("%6,d: %6,d = %s".fmt(n, ds.sum(0), ds.sort().concat(" + ")))
}
}
```
```zkl
oaw:=oddAbundants();
println("First 25 abundant odd numbers:");
oaw.walk(25) : printOAs(_);
println("\nThe one thousandth abundant odd number is:");
oaw.drop(1_000 - 25).value : printOAs(_);
println("\nThe first abundant odd number above one billion is:");
printOAs(oddAbundants(1_000_000_000).next());
```
{{out}}
945: 975 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315
1,575: 1,649 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525
2,205: 2,241 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735
2,835: 2,973 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945
3,465: 4,023 = 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155
4,095: 4,641 = 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365
4,725: 5,195 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575
5,355: 5,877 = 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785
5,775: 6,129 = 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925
5,985: 6,495 = 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995
6,435: 6,669 = 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145
6,615: 7,065 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205
6,825: 7,063 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275
7,245: 7,731 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415
7,425: 7,455 = 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475
7,875: 8,349 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625
8,085: 8,331 = 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695
8,415: 8,433 = 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805
8,505: 8,967 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835
8,925: 8,931 = 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975
9,135: 9,585 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045
9,555: 9,597 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185
9,765: 10,203 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255
10,395: 12,645 = 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465
11,025: 11,946 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675
The one thousandth abundant odd number is:
492,975: 519,361 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325
The first abundant odd number above one billion is:
1,000,000,575: 1,083,561,009 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525
```