⚠️ Warning: This is a draft ⚠️
This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.
If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.
{{task}}
Calculate the sequence where each term an is the '''smallest natural number''' that has exactly '''n''' divisors.
;Task Show here, on this page, at least the first '''15''' terms of the sequence.
;See also
:*[[oeis:A005179|OEIS:A005179]]
;Related tasks
:[[Sequence: smallest number greater than previous term with exactly n divisors]] :[[Sequence: nth number with exactly n divisors]]
ALGOL 68
{{Trans|C}}
BEGIN
PROC count divisors = ( INT n )INT:
BEGIN
INT count := 0;
FOR i WHILE i*i <= n DO
IF n MOD i = 0 THEN
count +:= IF i = n OVER i THEN 1 ELSE 2 FI
FI
OD;
count
END # count divisors # ;
INT max = 15;
[ max ]INT seq;FOR i TO max DO seq[ i ] := 0 OD;
INT found := 0;
FOR i WHILE found < max DO
IF INT divisors = count divisors( i );
divisors <= max
THEN
# have an i with an appropriate number of divisors #
IF seq[ divisors ] = 0 THEN
# this is the first i with that many divisors #
seq[ divisors ] := i;
found +:= 1
FI
FI
OD;
print( ( "The first ", whole( max, 0 ), " terms of the sequence are:", newline ) );
FOR i TO max DO
print( ( whole( seq( i ), 0 ), " " ) )
OD;
print( ( newline ) )
END
{{out}}
The first 15 terms of the sequence are:
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
AWK
# syntax: GAWK -f SEQUENCE_SMALLEST_NUMBER_WITH_EXACTLY_N_DIVISORS.AWK
# converted from Kotlin
BEGIN {
limit = 15
printf("first %d terms:",limit)
i = 1
n = 0
while (n < limit) {
k = count_divisors(i)
if (k <= limit && seq[k-1]+0 == 0) {
seq[k-1] = i
n++
}
i++
}
for (i=0; i<limit; i++) {
printf(" %d",seq[i])
}
printf("\n")
exit(0)
}
function count_divisors(n, count,i) {
for (i=1; i*i<=n; i++) {
if (n % i == 0) {
count += (i == n / i) ? 1 : 2
}
}
return(count)
}
{{out}}
first 15 terms: 1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
C
{{trans|Go}}
#include <stdio.h> #define MAX 15 int count_divisors(int n) { int i, count = 0; for (i = 1; i * i <= n; ++i) { if (!(n % i)) { if (i == n / i) count++; else count += 2; } } return count; } int main() { int i, k, n, seq[MAX]; for (i = 0; i < MAX; ++i) seq[i] = 0; printf("The first %d terms of the sequence are:\n", MAX); for (i = 1, n = 0; n < MAX; ++i) { k = count_divisors(i); if (k <= MAX && seq[k - 1] == 0) { seq[k - 1] = i; ++n; } } for (i = 0; i < MAX; ++i) printf("%d ", seq[i]); printf("\n"); return 0; }
{{out}}
The first 15 terms of the sequence are:
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
C++
{{trans|C}}
#include <iostream> #define MAX 15 using namespace std; int count_divisors(int n) { int count = 0; for (int i = 1; i * i <= n; ++i) { if (!(n % i)) { if (i == n / i) count++; else count += 2; } } return count; } int main() { int i, k, n, seq[MAX]; for (i = 0; i < MAX; ++i) seq[i] = 0; cout << "The first " << MAX << " terms of the sequence are:" << endl; for (i = 1, n = 0; n < MAX; ++i) { k = count_divisors(i); if (k <= MAX && seq[k - 1] == 0) { seq[k - 1] = i; ++n; } } for (i = 0; i < MAX; ++i) cout << seq[i] << " "; cout << endl; return 0; }
{{out}}
The first 15 terms of the sequence are:
1 2 4 6 16 12 64 24 36 48 1024 60 0 192 144
=={{header|F_Sharp|F#}}== This task uses [http://www.rosettacode.org/wiki/Extensible_prime_generator#The_function Extensible Prime Generator (F#)]
// Find Antı-Primes plus. Nigel Galloway: April 9th., 2019 // Increasing the value 14 will increase the number of anti-primes plus found let fI=primes|>Seq.take 14|>Seq.map bigint|>List.ofSeq let N=Seq.reduce(*) fI let fG g=Seq.unfold(fun ((n,i,e) as z)->Some(z,(n+1,i+1,(e*g)))) (1,2,g) let fE n i=n|>Seq.collect(fun(n,e,g)->Seq.map(fun(a,c,b)->(a,c*e,g*b)) (i|>Seq.takeWhile(fun(g,_,_)->g<=n))|> Seq.takeWhile(fun(_,_,n)->n<N)) let fL=let mutable g=0 in (fun n->g<-g+1; n=g) let n=Seq.concat(Seq.scan(fun n g->fE n (fG g)) (seq[(2147483647,1,1I)]) fI)|>List.ofSeq|>List.groupBy(fun(_,n,_)->n)|>List.sortBy(fun(n,_)->n)|>List.takeWhile(fun(n,_)->fL n) for n,g in n do printfn "%d->%A" n (g|>List.map(fun(_,_,n)->n)|>List.min)
{{out}}
1->1
2->2
3->4
4->6
5->16
6->12
7->64
8->24
9->36
10->48
11->1024
12->60
13->4096
14->192
15->144
16->120
17->65536
18->180
19->262144
20->240
21->576
22->3072
23->4194304
24->360
25->1296
26->12288
27->900
28->960
29->268435456
30->720
31->1073741824
32->840
33->9216
34->196608
35->5184
36->1260
37->68719476736
38->786432
39->36864
40->1680
41->1099511627776
42->2880
43->4398046511104
44->15360
45->3600
46->12582912
47->70368744177664
48->2520
49->46656
50->6480
51->589824
52->61440
53->4503599627370496
54->6300
55->82944
56->6720
57->2359296
58->805306368
Real: 00:00:01.079, CPU: 00:00:01.080, GC gen0: 47, gen1: 0
Factor
USING: fry kernel lists lists.lazy math math.primes.factors
prettyprint sequences ;
: A005179 ( -- list )
1 lfrom [
1 swap '[ dup divisors length _ = ] [ 1 + ] until
] lmap-lazy ;
15 A005179 ltake list>array .
{{out}}
{ 1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144 }
Go
package main import "fmt" func countDivisors(n int) int { count := 0 for i := 1; i*i <= n; i++ { if n%i == 0 { if i == n/i { count++ } else { count += 2 } } } return count } func main() { const max = 15 seq := make([]int, max) fmt.Println("The first", max, "terms of the sequence are:") for i, n := 1, 0; n < max; i++ { if k := countDivisors(i); k <= max && seq[k-1] == 0 { seq[k-1] = i n++ } } fmt.Println(seq) }
{{out}}
The first 15 terms of the sequence are:
[1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144]
Java
{{trans|C}}
import java.util.Arrays; public class OEIS_A005179 { static int count_divisors(int n) { int count = 0; for (int i = 1; i * i <= n; ++i) { if (n % i == 0) { if (i == n / i) count++; else count += 2; } } return count; } public static void main(String[] args) { final int max = 15; int[] seq = new int[max]; System.out.printf("The first %d terms of the sequence are:\n", max); for (int i = 1, n = 0; n < max; ++i) { int k = count_divisors(i); if (k <= max && seq[k - 1] == 0) { seq[k- 1] = i; n++; } } System.out.println(Arrays.toString(seq)); } }
{{out}}
The first 15 terms of the sequence are:
[1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144]
Julia
{{works with|Julia|1.2}}
using Primes numfactors(n) = reduce(*, e+1 for (_,e) in factor(n); init=1) A005179(n) = findfirst(k -> numfactors(k) == n, 1:typemax(Int)) println("The first 15 terms of the sequence are:") println(map(A005179, 1:15))
{{out}}
First 15 terms of OEIS sequence A005179:
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
Kotlin
{{trans|Go}}
// Version 1.3.21 const val MAX = 15 fun countDivisors(n: Int): Int { var count = 0 var i = 1 while (i * i <= n) { if (n % i == 0) { count += if (i == n / i) 1 else 2 } i++ } return count } fun main() { var seq = IntArray(MAX) println("The first $MAX terms of the sequence are:") var i = 1 var n = 0 while (n < MAX) { var k = countDivisors(i) if (k <= MAX && seq[k - 1] == 0) { seq[k - 1] = i n++ } i++ } println(seq.asList()) }
{{output}}
The first 15 terms of the sequence are:
[1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144]
Perl
{{libheader|ntheory}}
use strict; use warnings; use ntheory 'divisors'; print "First 15 terms of OEIS: A005179\n"; for my $n (1..15) { my $l = 0; while (++$l) { print "$l " and last if $n == divisors($l); } }
{{out}}
First 15 terms of OEIS: A005179
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
Perl 6
{{works with|Rakudo|2019.03}}
sub div-count (\x) {
return 2 if x.is-prime;
+flat (1 .. x.sqrt.floor).map: -> \d {
unless x % d { my \y = x div d; y == d ?? y !! (y, d) }
}
}
my $limit = 15;
put "First $limit terms of OEIS:A005179";
put (1..$limit).map: -> $n { first { $n == .&div-count }, 1..Inf };
{{out}}
First 15 terms of OEIS:A005179
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
Phix
naive
constant limit = 15
sequence res = repeat(0,limit)
integer found = 0, n = 1
while found<limit do
integer k = length(factors(n,1))
if k<=limit and res[k]=0 then
res[k] = n
found += 1
end if
n += 1
end while
printf(1,"The first %d terms are: %v\n",{limit,res})
{{out}}
The first 15 terms are: {1,2,4,6,16,12,64,24,36,48,1024,60,4096,192,144}
You would need something quite a bit smarter to venture over a limit of 28.
advanced
Using the various formula from the OEIS:A005179 link above.
get_primes() and product() have recently been added as new builtins, if necessary see [[Extensible_prime_generator#Phix|Extensible_prime_generator]] and [[Deconvolution/2D%2B#Phix]].
constant limit = iff(machine_bits()=32?58:66)
sequence found = repeat(0,limit)
integer n = 1
procedure populate_found(integer i)
while found[i]=0 do
integer k = length(factors(n,1))
if k<=limit and found[k]=0 then
found[k] = n
end if
n += 1
end while
end procedure
for i=1 to limit do
sequence f = factors(i,1)
integer lf = length(f)
atom ri
if lf<=2 then ri = power(2,i-1) -- prime (or 1)
elsif lf=3 then ri = power(6,f[2]-1) -- p^2 (eg f={1,5,25})
elsif f[2]>2 -- (see note)
and f[$] = power(f[2],lf-1) then ri = power(product(get_primes(-(lf-1))),f[2]-1) -- p^k (eg f={1,3,9,27})
elsif lf=4 then ri = power(2,f[3]-1)*power(3,f[2]-1) -- p*q (eg f={1,2,3,6})
else populate_found(i) ri = found[i] -- do the rest manually
end if
printf(1,"%d->%d\n",{i,ri})
end for
Note: the f[2]>2 test should really be something more like >log(get_primes(-(lf-1))[$])/log(2), apparently, but everything seems ok within the IEEE 754 53/64 bit limits this imposes. It takes longer, afaict, to print the answers than it did to calculate them, tee hee! {{out}} 64-bit (as shown) manages 8 more answers than 32-bit, which as per limit halts on 58: on 32 bit the accuracy limit is 2^53, hence the result for 59, which is 2^58, would get printed wrong since the first /10 needed to print it rounds to the nearest 16 or so. It is quite probably perfectly accurate internally up to much higher limits, but proving/showing that is a bit of a problem, which would in turn probably be easiest to solve by simply rewriting this to use gmp/mpir.
1->1
2->2
3->4
4->6
5->16
6->12
7->64
8->24
9->36
10->48
11->1024
12->60
13->4096
14->192
15->144
16->120
17->65536
18->180
19->262144
20->240
21->576
22->3072
23->4194304
24->360
25->1296
26->12288
27->900
28->960
29->268435456
30->720
31->1073741824
32->840
33->9216
34->196608
35->5184
36->1260
37->68719476736
38->786432
39->36864
40->1680
41->1099511627776
42->2880
43->4398046511104
44->15360
45->3600
46->12582912
47->70368744177664
48->2520
49->46656
50->6480
51->589824
52->61440
53->4503599627370496
54->6300
55->82944
56->6720
57->2359296
58->805306368
59->288230376151711744
60->5040
61->1152921504606846976
62->3221225472
63->14400
64->7560
65->331776
66->46080
insane
A rather silly (but successful) attempt to reverse engineer all the rules up to 2000.
I got it down to just 11 of them, with only 1 being a complete fudge. Obviously, the fewer cases each covers, the less sound it is, and those mini-tables for np/p2/p3/p5 and adj are not exactly, um, scientific. Completes in about 0.1s {{libheader|mpfr}}
include mpfr.e
mpz r = mpz_init(),
pn = mpz_init()
sequence rule_names = {},
rule_counts = {}
for i=1 to 2000 do
sequence pf = prime_factors(i,true), ri, adj
integer lf = length(pf), np, p2, p3, p5, p, e
string what
if lf>10 then ?9/0 end if
if lf<=1 then what = "prime (proper rule)"
np = 1
adj = {i}
elsif pf[$]=2 then what = "2^k (made up rule)"
np = lf-1
p2 = {2,4,4,4,4,4,4,8,8}[np]
p3 = {2,2,2,2,4,4,4,4,4}[np]
np = {2,2,3,4,4,5,6,6,7}[np]
adj = {p2,p3}
elsif pf[$]=3
and pf[$-1]=2 then what = "2^k*3 (made up rule)"
np = lf-1
p2 = {3,3,4,4,4,6,6,6,6}[np]
p3 = {2,2,3,3,3,4,4,4,4}[np]
np = {2,3,3,4,5,5,6,7,8}[np]
adj = {p2,p3}
elsif lf>4
and pf[$-1]=2 then what="2^k*p (made up rule)"
np = lf-1
adj = {0,4}
elsif lf>4
and pf[$]=3
and pf[$-1]=3
and pf[$-2]=2 then what="2^k*3^2*p (made up rule)"
np = lf-4
p3 = {3,3,3,4,4}[np]
p5 = {2,2,2,3,3}[np]
np = {4,5,6,6,7}[np]
adj = {6,p3,p5}
elsif lf>4
and pf[$]=3
and pf[$-2]=3
and pf[$-4]=2 then what="2^k*3^3*p (made up rule)"
np = lf-1
adj = {6}
elsif lf>5
and pf[$]>3
and pf[$-1]=3
and pf[$-4]=3
and pf[2]=3
and (pf[1]=2 or pf[$]>5) then what="2^k*3^4*p (made up rule)"
np = lf
adj = {}
elsif lf>4
and pf[$-1]=3
and pf[$-4]=3
and (lf>5 or pf[$]=3) then what="[2^k]*3^(>=4)*p (made up rule)"
np = lf-1
adj = {9,pf[$]}&reverse(pf[1..$-3]) -- <bsg>
elsif lf>=7
and pf[$]>3
and pf[$-1]=3
and pf[$-2]=2 then what="2^k*3*p (made up rule)"
np = lf-1
adj = {0,4,3}
elsif i=1440
and pf={2,2,2,2,2,3,3,5} then what="1440 (complete fudge)"
-- nothing quite like this, nothing to build any pattern from...
np = 7
adj = {6,5,3,2,2,2,2}
else what="general (proper rule)"
-- (note this incorporates the p^2, (p>2)^k, p*q, and p*m*q rules)
np = lf
adj = {}
end if
ri = get_primes(-np)
for j=1 to length(adj) do
integer aj = adj[j]
if aj!=0 then pf[-j] = aj end if
end for
for j=1 to np do
ri[j] = {ri[j],pf[-j]-1}
end for
string short = "" -- (eg "2^2*3^3" form)
mpz_set_si(r,1) -- (above as big int)
for j=1 to length(ri) do
{p, e} = ri[j]
if length(short) then short &= "*" end if
short &= sprintf("%d",p)
if e!=1 then
short &= sprintf("^%d",{e})
end if
mpz_ui_pow_ui(pn,p,e)
mpz_mul(r,r,pn)
end for
if i<=15 or remainder(i-1,250)>=248 or i=1440 then
string rs = mpz_get_str(r)
if length(rs)>20 then
rs[6..-6] = sprintf("<-- %d digits -->",length(rs)-10)
end if
if short="2^0" then short = "1" end if
printf(1,"%4d : %25s %30s %s\n",{i,short,rs,what})
end if
integer k = find(what,rule_names)
if k=0 then
rule_names = append(rule_names,what)
rule_counts = append(rule_counts,1)
else
rule_counts[k] += 1
end if
end for
integer lr = length(rule_names)
printf(1,"\nrules(%d):\n",lr)
sequence tags = custom_sort(rule_counts, tagset(lr))
for i=1 to lr do
integer ti = tags[-i]
printf(1," %30s:%d\n",{rule_names[ti],rule_counts[ti]})
end for
{r,pn} = mpz_free({r,pn})
{{out}}
1 : 1 1 prime (proper rule)
2 : 2 2 prime (proper rule)
3 : 2^2 4 prime (proper rule)
4 : 2*3 6 2^k (made up rule)
5 : 2^4 16 prime (proper rule)
6 : 2^2*3 12 2^k*3 (made up rule)
7 : 2^6 64 prime (proper rule)
8 : 2^3*3 24 2^k (made up rule)
9 : 2^2*3^2 36 general (proper rule)
10 : 2^4*3 48 general (proper rule)
11 : 2^10 1024 prime (proper rule)
12 : 2^2*3*5 60 2^k*3 (made up rule)
13 : 2^12 4096 prime (proper rule)
14 : 2^6*3 192 general (proper rule)
15 : 2^4*3^2 144 general (proper rule)
249 : 2^82*3^2 43521<-- 16 digits -->22336 general (proper rule)
250 : 2^4*3^4*5^4*7 5670000 general (proper rule)
499 : 2^498 81834<-- 140 digits -->97344 prime (proper rule)
500 : 2^4*3^4*5^4*7*11 62370000 general (proper rule)
749 : 2^106*3^6 59143<-- 25 digits -->22656 general (proper rule)
750 : 2^4*3^4*5^4*7^2*11 436590000 general (proper rule)
999 : 2^36*3^2*5^2*7^2 757632231014400 general (proper rule)
1000 : 2^4*3^4*5^4*7*11*13 810810000 general (proper rule)
1249 : 2^1248 48465<-- 366 digits -->22656 prime (proper rule)
1250 : 2^4*3^4*5^4*7^4*11 21392910000 general (proper rule)
1440 : 2^5*3^4*5^2*7*11*13*17 1102701600 1440 (complete fudge)
1499 : 2^1498 87686<-- 441 digits -->37344 prime (proper rule)
1500 : 2^4*3^4*5^4*7^2*11*13 5675670000 general (proper rule)
1749 : 2^52*3^10*5^2 66483<-- 12 digits -->57600 general (proper rule)
1750 : 2^6*3^4*5^4*7^4*11 85571640000 general (proper rule)
1999 : 2^1998 28703<-- 592 digits -->57344 prime (proper rule)
2000 : 2^4*3^4*5^4*7*11*13*17 13783770000 general (proper rule)
rules(11):
general (proper rule):1583
prime (proper rule):304
2^k*p (made up rule):59
2^k*3*p (made up rule):9
2^k*3^3*p (made up rule):9
2^k (made up rule):9
2^k*3 (made up rule):9
[2^k]*3^(>=4)*p (made up rule):8
2^k*3^2*p (made up rule):5
2^k*3^4*p (made up rule):4
1440 (complete fudge):1
Python
def divisors(n): divs = [1] for ii in range(2, int(n ** 0.5) + 3): if n % ii == 0: divs.append(ii) divs.append(int(n / ii)) divs.append(n) return list(set(divs)) def sequence(max_n=None): n = 0 while True: n += 1 ii = 0 if max_n is not None: if n > max_n: break while True: ii += 1 if len(divisors(ii)) == n: yield ii break if __name__ == '__main__': for item in sequence(15): print(item)
Output:
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144
REXX
/*REXX program finds and displays the smallest number with exactly N divisors.*/
parse arg N . /*obtain optional argument from the CL.*/
if N=='' | N=="," then N= 15 /*Not specified? Then use the default.*/
say '──divisors── ──smallest number with N divisors──' /*display title for the numbers.*/
@.= /*the @ array is used for memoization*/
do i=1 for N /*step through a number of divisors. */
do j=1+(i\==1) by 1+(i\==1) /*now, search for a number that ≡ #divs*/
if @.j==. then iterate /*has this number already been found? */
d= #divs(j); if d\==i then iterate /*get # divisors; Is not equal? Skip.*/
say center(i, 12) right(j, 19) /*display the #divs and the smallest #.*/
@.j=. /*mark as having found #divs for this J*/
leave /*found a number, so now get the next I*/
end /*j*/
end /*i*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
#divs: procedure; parse arg x 1 y /*X and Y: both set from 1st argument.*/
if x<7 then do /*handle special cases for numbers < 7.*/
if x<3 then return x /* " " " " one and two.*/
if x<5 then return x - 1 /* " " " " three & four*/
if x==5 then return 2 /* " " " " five. */
if x==6 then return 4 /* " " " " six. */
end
odd= x // 2 /*check if X is odd or not. */
if odd then do; #= 1; end /*Odd? Assume Pdivisors count of 1.*/
else do; #= 3; y= x%2; end /*Even? " " " " 3.*/
/* [↑] start with known num of Pdivs.*/
do k=3 for x%2-3 by 1+odd while k<y /*for odd numbers, skip evens.*/
if x//k==0 then do /*if no remainder, then found a divisor*/
#=#+2; y=x%k /*bump # Pdivs, calculate limit Y. */
if k>=y then do; #= #-1; leave; end /*limit?*/
end /* ___ */
else if k*k>x then leave /*only divide up to √ x */
end /*k*/ /* [↑] this form of DO loop is faster.*/
return #+1 /*bump "proper divisors" to "divisors".*/
{{out|output|text= when using the default input:}}
──divisors── ──smallest number with N divisors──
1 1
2 2
3 4
4 6
5 16
6 12
7 64
8 24
9 36
10 48
11 1024
12 60
13 4096
14 192
15 144
Ruby
require 'prime' def num_divisors(n) n.prime_division.inject(1){|prod, (_p,n)| prod *= (n + 1) } end def first_with_num_divs(n) (1..).detect{|i| num_divisors(i) == n } end p (1..15).map{|n| first_with_num_divs(n) }
{{out}}
[1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144]
Sidef
func n_divisors(n) { 1..Inf -> first_by { .sigma0 == n } } say 15.of { n_divisors(_+1) }
{{out}}
[1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144]
zkl
fcn countDivisors(n)
{ [1.. n.toFloat().sqrt()].reduce('wrap(s,i){ s + (if(0==n%i) 1 + (i!=n/i)) },0) }
A005179w:=(1).walker(*).tweak(fcn(n){
var N=0,cache=Dictionary();
if(cache.find(n)) return(cache.pop(n)); // prune
while(1){
if(n == (d:=countDivisors(N+=1))) return(N);
if(n<d and not cache.find(d)) cache[d]=N;
}
});
N:=15;
println("First %d terms of OEIS:A005179".fmt(N));
A005179w.walk(N).concat(" ").println();
{{out}}
First 15 terms of OEIS:A005179
1 2 4 6 16 12 64 24 36 48 1024 60 4096 192 144