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{{task}}
The sequence is from the natural numbers and is defined by:
a(1) = 1;a(2) = Start = 2;- for n > 2,
a(n)shares at least one prime factor witha(n-1)and is the ''smallest'' such natural number ''not already used''.
The sequence is called the [http://oeis.org/A064740 EKG sequence] (after its visual similarity to an electrocardiogram when graphed).
Variants of the sequence can be generated starting 1, N where N is any natural number larger than one. For the purposes of this task let us call:
- The sequence described above , starting
1, 2, ...theEKG(2)sequence; - the sequence starting
1, 3, ...theEKG(3)sequence; - ... the sequence starting
1, N, ...theEKG(N)sequence.
;Convergence If an algorithm that keeps track of the minimum amount of numbers and their corresponding prime factors used to generate the next term is used, then this may be known as the generators essential '''state'''. Two EKG generators with differing starts can converge to produce the same sequence after initial differences.
EKG(N1) and EKG(N2) are said to to have converged at and after generation a(c) if state_of(EKG(N1).a(c)) == state_of(EKG(N2).a(c)).
;Task:
Calculate and show here the first 10 members of EKG(2).
Calculate and show here the first 10 members of EKG(5).
Calculate and show here the first 10 members of EKG(7).
Calculate and show here the first 10 members of EKG(9).
Calculate and show here the first 10 members of EKG(10).
Calculate and show here at which term EKG(5) and EKG(7) converge ('''stretch goal''').
;Related Tasks:
[[Greatest common divisor]]
[[Sieve of Eratosthenes]]
;Reference:
- [https://www.youtube.com/watch?v=yd2jr30K2R4 The EKG Sequence and the Tree of Numbers]. (Video).
C
{{trans|Go}}
#include <stdio.h> #include <stdlib.h> #define TRUE 1 #define FALSE 0 #define LIMIT 100 typedef int bool; int compareInts(const void *a, const void *b) { int aa = *(int *)a; int bb = *(int *)b; return aa - bb; } bool contains(int a[], int b, size_t len) { int i; for (i = 0; i < len; ++i) { if (a[i] == b) return TRUE; } return FALSE; } int gcd(int a, int b) { while (a != b) { if (a > b) a -= b; else b -= a; } return a; } bool areSame(int s[], int t[], size_t len) { int i; qsort(s, len, sizeof(int), compareInts); qsort(t, len, sizeof(int), compareInts); for (i = 0; i < len; ++i) { if (s[i] != t[i]) return FALSE; } return TRUE; } int main() { int s, n, i; int starts[5] = {2, 5, 7, 9, 10}; int ekg[5][LIMIT]; for (s = 0; s < 5; ++s) { ekg[s][0] = 1; ekg[s][1] = starts[s]; for (n = 2; n < LIMIT; ++n) { for (i = 2; ; ++i) { // a potential sequence member cannot already have been used // and must have a factor in common with previous member if (!contains(ekg[s], i, n) && gcd(ekg[s][n - 1], i) > 1) { ekg[s][n] = i; break; } } } printf("EKG(%2d): [", starts[s]); for (i = 0; i < 30; ++i) printf("%d ", ekg[s][i]); printf("\b]\n"); } // now compare EKG5 and EKG7 for convergence for (i = 2; i < LIMIT; ++i) { if (ekg[1][i] == ekg[2][i] && areSame(ekg[1], ekg[2], i)) { printf("\nEKG(5) and EKG(7) converge at term %d\n", i + 1); return 0; } } printf("\nEKG5(5) and EKG(7) do not converge within %d terms\n", LIMIT); return 0; }
{{out}}
EKG( 2): [1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 33 27 30 25 35 28 26 13 39 36]
EKG( 5): [1 5 10 2 4 6 3 9 12 8 14 7 21 15 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG( 7): [1 7 14 2 4 6 3 9 12 8 10 5 15 18 16 20 22 11 33 21 24 26 13 39 27 30 25 35 28 32]
EKG( 9): [1 9 3 6 2 4 8 10 5 15 12 14 7 21 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG(10): [1 10 2 4 6 3 9 12 8 14 7 21 15 5 20 16 18 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG(5) and EKG(7) converge at term 21
=={{header|F_Sharp|F#}}==
The Function
This task uses [http://www.rosettacode.org/wiki/Extensible_prime_generator#The_function Extensible Prime Generator (F#)]
// Generate EKG Sequences. Nigel Galloway: December 6th., 2018 let EKG n=seq{ let fN,fG=let i=System.Collections.Generic.Dictionary<int,int>() let fN g=(if not (i.ContainsKey g) then i.[g]<-g);(g,i.[g]) ((fun e->i.[e]<-i.[e]+e), (fun l->l|>List.map fN)) let fU l= pCache|>Seq.takeWhile(fun n->n<=l)|>Seq.filter(fun n->l%n=0)|>List.ofSeq let rec EKG l (α,β)=seq{let b=fU β in if (β=n||β<snd((fG b|>List.maxBy snd))) then fN α; yield! EKG l (fG l|>List.minBy snd) else fN α;yield β;yield! EKG b (fG b|>List.minBy snd)} yield! seq[1;n]; let g=fU n in yield! EKG g (fG g|>Seq.minBy snd)} let EKGconv n g=Seq.zip(EKG n)(EKG g)|>Seq.skip 2|>Seq.scan(fun(n,i,g,e)(l,β)->(Set.add l n,Set.add β i,l,β))(set[1;n],set[1;g],0,0)|>Seq.takeWhile(fun(n,i,g,e)->g<>e||n<>i)
The Task
EKG 2 |> Seq.take 45 |> Seq.iter(printf "%2d, ") EKG 3 |> Seq.take 45 |> Seq.iter(printf "%2d, ") EKG 5 |> Seq.take 45 |> Seq.iter(printf "%2d, ") EKG 7 |> Seq.take 45 |> Seq.iter(printf "%2d, ") EKG 9 |> Seq.take 45 |> Seq.iter(printf "%2d, ") EKG 10 |> Seq.take 45 |> Seq.iter(printf "%2d, ") printfn "%d" (let n,_,_,_=EKGconv 2 5|>Seq.last in ((Set.count n)+1)
{{out}}
1, 2, 4, 6, 3, 9,12, 8,10, 5,15,18,14, 7,21,24,16,20,22,11,33,27,30,25,35,28,26,13,39,36,32,34,17,51,42,38,19,57,45,40,44,46,23,69,48
1, 3, 6, 2, 4, 8,10, 5,15, 9,12,14, 7,21,18,16,20,22,11,33,24,26,13,39,27,30,25,35,28,32,34,17,51,36,38,19,57,42,40,44,46,23,69,45,48
1, 5,10, 2, 4, 6, 3, 9,12, 8,14, 7,21,15,18,16,20,22,11,33,24,26,13,39,27,30,25,35,28,32,34,17,51,36,38,19,57,42,40,44,46,23,69,45,48
45
Extra Credit
prıntfn "%d" (EKG 2|>Seq.takeWhile(fun n->n<>104729) ((Seq.length n)+1)
{{out}}
203786
Real: 00:10:21.967, CPU: 00:10:25.300, GC gen0: 65296, gen1: 1
Go
package main import ( "fmt" "sort" ) func contains(a []int, b int) bool { for _, j := range a { if j == b { return true } } return false } func gcd(a, b int) int { for a != b { if a > b { a -= b } else { b -= a } } return a } func areSame(s, t []int) bool { le := len(s) if le != len(t) { return false } sort.Ints(s) sort.Ints(t) for i := 0; i < le; i++ { if s[i] != t[i] { return false } } return true } func main() { const limit = 100 starts := [5]int{2, 5, 7, 9, 10} var ekg [5][limit]int for s, start := range starts { ekg[s][0] = 1 ekg[s][1] = start for n := 2; n < limit; n++ { for i := 2; ; i++ { // a potential sequence member cannot already have been used // and must have a factor in common with previous member if !contains(ekg[s][:n], i) && gcd(ekg[s][n-1], i) > 1 { ekg[s][n] = i break } } } fmt.Printf("EKG(%2d): %v\n", start, ekg[s][:30]) } // now compare EKG5 and EKG7 for convergence for i := 2; i < limit; i++ { if ekg[1][i] == ekg[2][i] && areSame(ekg[1][:i], ekg[2][:i]) { fmt.Println("\nEKG(5) and EKG(7) converge at term", i+1) return } } fmt.Println("\nEKG5(5) and EKG(7) do not converge within", limit, "terms") }
{{out}}
EKG( 2): [1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 33 27 30 25 35 28 26 13 39 36]
EKG( 5): [1 5 10 2 4 6 3 9 12 8 14 7 21 15 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG( 7): [1 7 14 2 4 6 3 9 12 8 10 5 15 18 16 20 22 11 33 21 24 26 13 39 27 30 25 35 28 32]
EKG( 9): [1 9 3 6 2 4 8 10 5 15 12 14 7 21 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG(10): [1 10 2 4 6 3 9 12 8 14 7 21 15 5 20 16 18 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG(5) and EKG(7) converge at term 21
Haskell
import Data.List (findIndex, isPrefixOf, tails) import Data.Maybe (fromJust) seqEKGRec :: Int -> Int -> [Int] -> [Int] seqEKGRec _ 0 l = l seqEKGRec k n [] = seqEKGRec k (n - 2) [k, 1] seqEKGRec k n l@(h:t) = seqEKGRec k (n - 1) (head (filter (\i -> notElem i l && (gcd h i > 1)) [2 ..]) : l) seqEKG :: Int -> Int -> [Int] seqEKG k n = reverse (seqEKGRec k n []) main :: IO () main = mapM_ (\x -> putStr "EKG (" >> (putStr . show $ x) >> putStr ") is " >> print (seqEKG x 20)) [2, 5, 7, 9, 10] >> putStr "EKG(5) and EKG(7) converge at " >> print ((+ 1) $ fromJust $ findIndex (isPrefixOf (replicate 20 True)) (tails (zipWith (==) (seqEKG 7 80) (seqEKG 5 80))))
{{out}}
EKG (2) is [1,2,4,6,3,9,12,8,10,5,15,18,14,7,21,24,16,20,22,11]
EKG (5) is [1,5,10,2,4,6,3,9,12,8,14,7,21,15,18,16,20,22,11,33]
EKG (7) is [1,7,14,2,4,6,3,9,12,8,10,5,15,18,16,20,22,11,33,21]
EKG (9) is [1,9,3,6,2,4,8,10,5,15,12,14,7,21,18,16,20,22,11,33]
EKG (10) is [1,10,2,4,6,3,9,12,8,14,7,21,15,5,20,16,18,22,11,33]
EKG(5) and EKG(7) converge at 21
J
Until =: 2 :'u^:(0-:v)^:_' NB. unused but so fun
prime_factors_of_tail =: ~.@:q:@:{:
numbers_not_in_list =: -.~ >:@:i.@:(>./)
ekg =: 3 :0 NB. return next sequence
if. 1 = # y do. NB. initialize
1 , y
return.
end.
a =. prime_factors_of_tail y
b =. numbers_not_in_list y
index_of_lowest =. {. _ ,~ I. 1 e."1 a e."1 q:b
if. index_of_lowest < _ do. NB. if the list doesn't need extension
y , index_of_lowest { b
return.
end.
NB. otherwise extend the list
b =. >: >./ y
while. 1 -.@:e. a e. q: b do.
b =. >: b
end.
y , b
)
ekg^:9&>2 5 7 9 10
1 2 4 6 3 9 12 8 10 5
1 5 10 2 4 6 3 9 12 8
1 7 14 2 4 6 3 9 12 8
1 9 3 6 2 4 8 10 5 15
1 10 2 4 6 3 9 12 8 14
assert 9 -: >:Until(>&8) 2
assert (,2) -: prime_factors_of_tail 6 8 NB. (nub of)
assert 3 4 5 -: numbers_not_in_list 1 2 6
Somewhat shorter is ekg2,
index_of_lowest =: [: {. _ ,~ [: I. 1 e."1 prime_factors_of_tail e."1 q:@:numbers_not_in_list
g =: 3 :0 NB. return sequence with next term appended
a =. prime_factors_of_tail y
(, (index_of_lowest { numbers_not_in_list)`(([: >:Until(1 e. a e. q:) [: >: >./))@.(_ = index_of_lowest)) y
)
ekg2 =: (1&,)`g@.(1<#)
assert (3 -: index_of_lowest { numbers_not_in_list)1 2 4 6
assert (ekg^:9&> -: ekg2^:9&>) 2 5 7 9 10
Julia
{{trans|Perl}}
using Primes function ekgsequence(n, limit) ekg::Array{Int,1} = [1, n] while length(ekg) < limit for i in 2:2<<18 if all(j -> j != i, ekg) && gcd(ekg[end], i) > 1 push!(ekg, i) break end end end ekg end function convergeat(n, m, max = 100) ekgn = ekgsequence(n, max) ekgm = ekgsequence(m, max) for i in 3:max if ekgn[i] == ekgm[i] && sum(ekgn[1:i+1]) == sum(ekgm[1:i+1]) return i end end warn("no converge in $max terms") end [println(rpad("EKG($i): ", 9), join(ekgsequence(i, 30), " ")) for i in [2, 5, 7, 9, 10]] println("EKGs of 5 & 7 converge at term ", convergeat(5, 7))
{{output}}
EKG(2): 1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 33 27 30 25 35 28 26 13 39 36
EKG(5): 1 5 10 2 4 6 3 9 12 8 14 7 21 15 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32
EKG(7): 1 7 14 2 4 6 3 9 12 8 10 5 15 18 16 20 22 11 33 21 24 26 13 39 27 30 25 35 28 32
EKG(9): 1 9 3 6 2 4 8 10 5 15 12 14 7 21 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32
EKG(10): 1 10 2 4 6 3 9 12 8 14 7 21 15 5 20 16 18 22 11 33 24 26 13 39 27 30 25 35 28 32
EKGs of 5 & 7 converge at term 21
Kotlin
{{trans|Go}}
// Version 1.2.60 fun gcd(a: Int, b: Int): Int { var aa = a var bb = b while (aa != bb) { if (aa > bb) aa -= bb else bb -= aa } return aa } const val LIMIT = 100 fun main(args: Array<String>) { val starts = listOf(2, 5, 7, 9, 10) val ekg = Array(5) { IntArray(LIMIT) } for ((s, start) in starts.withIndex()) { ekg[s][0] = 1 ekg[s][1] = start for (n in 2 until LIMIT) { var i = 2 while (true) { // a potential sequence member cannot already have been used // and must have a factor in common with previous member if (!ekg[s].slice(0 until n).contains(i) && gcd(ekg[s][n - 1], i) > 1) { ekg[s][n] = i break } i++ } } System.out.printf("EKG(%2d): %s\n", start, ekg[s].slice(0 until 30)) } // now compare EKG5 and EKG7 for convergence for (i in 2 until LIMIT) { if (ekg[1][i] == ekg[2][i] && ekg[1].slice(0 until i).sorted() == ekg[2].slice(0 until i).sorted()) { println("\nEKG(5) and EKG(7) converge at term ${i + 1}") return } } println("\nEKG5(5) and EKG(7) do not converge within $LIMIT terms") }
{{output}}
EKG( 2): [1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, 33, 27, 30, 25, 35, 28, 26, 13, 39, 36]
EKG( 5): [1, 5, 10, 2, 4, 6, 3, 9, 12, 8, 14, 7, 21, 15, 18, 16, 20, 22, 11, 33, 24, 26, 13, 39, 27, 30, 25, 35, 28, 32]
EKG( 7): [1, 7, 14, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 16, 20, 22, 11, 33, 21, 24, 26, 13, 39, 27, 30, 25, 35, 28, 32]
EKG( 9): [1, 9, 3, 6, 2, 4, 8, 10, 5, 15, 12, 14, 7, 21, 18, 16, 20, 22, 11, 33, 24, 26, 13, 39, 27, 30, 25, 35, 28, 32]
EKG(10): [1, 10, 2, 4, 6, 3, 9, 12, 8, 14, 7, 21, 15, 5, 20, 16, 18, 22, 11, 33, 24, 26, 13, 39, 27, 30, 25, 35, 28, 32]
EKG(5) and EKG(7) converge at term 21
Perl
{{trans|Perl 6}}
use List::Util qw(none sum); sub gcd { my ($u,$v) = @_; $v ? gcd($v, $u%$v) : abs($u) } sub shares_divisors_with { gcd( $_[0], $_[1]) > 1 } sub EKG { my($n,$limit) = @_; my @ekg = (1, $n); while (@ekg < $limit) { for my $i (2..1e18) { next unless none { $_ == $i } @ekg and shares_divisors_with($ekg[-1], $i); push(@ekg, $i) and last; } } @ekg; } sub converge_at { my($n1,$n2) = @_; my $max = 100; my @ekg1 = EKG($n1,$max); my @ekg2 = EKG($n2,$max); do { return $_+1 if $ekg1[$_] == $ekg2[$_] && sum(@ekg1[0..$_]) == sum(@ekg2[0..$_])} for 2..$max; return "(no convergence in $max terms)"; } print "EKG($_): " . join(' ', EKG($_,10)) . "\n" for 2, 5, 7, 9, 10; print "EKGs of 5 & 7 converge at term " . converge_at(5, 7) . "\n"
{{out}}
EKG(2): 1 2 4 6 3 9 12 8 10 5
EKG(5): 1 5 10 2 4 6 3 9 12 8
EKG(7): 1 7 14 2 4 6 3 9 12 8
EKG(9): 1 9 3 6 2 4 8 10 5 15
EKG(10): 1 10 2 4 6 3 9 12 8 14
EKGs of 5 & 7 converge at term 21
Perl 6
{{works with|Rakudo Star|2018.04.1}}
sub infix:<shares-divisors-with> { ($^a gcd $^b) > 1 }
sub next-EKG ( *@s ) {
return first {
@s ∌ $_ and @s.tail shares-divisors-with $_
}, 2..*;
}
sub EKG ( Int $start ) { 1, $start, &next-EKG … * }
sub converge-at ( @ints ) {
my @ekgs = @ints.map: &EKG;
return (2 .. *).first: -> $i {
[==] @ekgs.map( *.[$i] ) and
[===] @ekgs.map( *.head($i).Set )
}
}
say "EKG($_): ", .&EKG.head(10) for 2, 5, 7, 9, 10;
for [5, 7], [2, 5, 7, 9, 10] -> @ints {
say "EKGs of (@ints[]) converge at term {$_+1}" with converge-at(@ints);
}
{{out}}
EKG(2): (1 2 4 6 3 9 12 8 10 5)
EKG(5): (1 5 10 2 4 6 3 9 12 8)
EKG(7): (1 7 14 2 4 6 3 9 12 8)
EKG(9): (1 9 3 6 2 4 8 10 5 15)
EKG(10): (1 10 2 4 6 3 9 12 8 14)
EKGs of (5 7) converge at term 21
EKGs of (2 5 7 9 10) converge at term 45
Phix
{{trans|C}}
constant LIMIT = 100
constant starts = {2, 5, 7, 9, 10}
sequence ekg = {}
string fmt = "EKG(%2d): ["&join(repeat("%d",min(LIMIT,30))," ")&"]\n"
for s=1 to length(starts) do
ekg = append(ekg,{1,starts[s]}&repeat(0,LIMIT-2))
for n=3 to LIMIT do
-- a potential sequence member cannot already have been used
-- and must have a factor in common with previous member
integer i = 2
while find(i,ekg[s])
or gcd(ekg[s][n-1],i)<=1 do
i += 1
end while
ekg[s][n] = i
end for
printf(1,fmt,starts[s]&ekg[s][1..min(LIMIT,30)])
end for
-- now compare EKG5 and EKG7 for convergence
constant EKG5 = find(5,starts),
EKG7 = find(7,starts)
string msg = sprintf("do not converge within %d terms", LIMIT)
for i=3 to LIMIT do
if ekg[EKG5][i]=ekg[EKG7][i]
and sort(ekg[EKG5][1..i-1])=sort(ekg[EKG7][1..i-1]) then
msg = sprintf("converge at term %d", i)
exit
end if
end for
printf(1,"\nEKG5(5) and EKG(7) %s\n", msg)
{{out}}
EKG( 2): [1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 33 27 30 25 35 28 26 13 39 36]
EKG( 5): [1 5 10 2 4 6 3 9 12 8 14 7 21 15 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG( 7): [1 7 14 2 4 6 3 9 12 8 10 5 15 18 16 20 22 11 33 21 24 26 13 39 27 30 25 35 28 32]
EKG( 9): [1 9 3 6 2 4 8 10 5 15 12 14 7 21 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG(10): [1 10 2 4 6 3 9 12 8 14 7 21 15 5 20 16 18 22 11 33 24 26 13 39 27 30 25 35 28 32]
EKG5(5) and EKG(7) converge at term 21
Python
Python: Using math.gcd
If this alternate definition of function EKG_gen is used then the output would be the same as above. Instead of keeping a cache of prime factors this calculates the gretest common divisor as needed.
from itertools import count, islice, takewhile from math import gcd def EKG_gen(start=2): """\ Generate the next term of the EKG together with the minimum cache of numbers left in its production; (the "state" of the generator). Using math.gcd """ c = count(start + 1) last, so_far = start, list(range(2, start)) yield 1, [] yield last, [] while True: for index, sf in enumerate(so_far): if gcd(last, sf) > 1: last = so_far.pop(index) yield last, so_far[::] break else: so_far.append(next(c)) def find_convergence(ekgs=(5,7)): "Returns the convergence point or zero if not found within the limit" ekg = [EKG_gen(n) for n in ekgs] for e in ekg: next(e) # skip initial 1 in each sequence return 2 + len(list(takewhile(lambda state: not all(state[0] == s for s in state[1:]), zip(*ekg)))) if __name__ == '__main__': for start in 2, 5, 7, 9, 10: print(f"EKG({start}):", str([n[0] for n in islice(EKG_gen(start), 10)])[1: -1]) print(f"\nEKG(5) and EKG(7) converge at term {find_convergence(ekgs=(5,7))}!")
{{out}} (Same as above).
EKG(2): 1, 2, 4, 6, 3, 9, 12, 8, 10, 5
EKG(5): 1, 5, 10, 2, 4, 6, 3, 9, 12, 8
EKG(7): 1, 7, 14, 2, 4, 6, 3, 9, 12, 8
EKG(9): 1, 9, 3, 6, 2, 4, 8, 10, 5, 15
EKG(10): 1, 10, 2, 4, 6, 3, 9, 12, 8, 14
EKG(5) and EKG(7) converge at term 21!
;Note: Despite EKG(5) and EKG(7) seeming to converge earlier, as seen above; their hidden states differ.
Here is those series out to 21 terms where you can see them diverge again before finally converging. The state is also shown.
# After running the above, in the terminal: from pprint import pprint as pp for start in 5, 7: print(f"EKG({start}):\n[(<next>, [<state>]), ...]") pp(([n for n in islice(EKG_gen(start), 21)]))
'''Generates:'''
EKG(5):
[(<next>, [<state>]), ...]
[(1, []),
(5, []),
(10, [2, 3, 4, 6, 7, 8, 9]),
(2, [3, 4, 6, 7, 8, 9]),
(4, [3, 6, 7, 8, 9]),
(6, [3, 7, 8, 9]),
(3, [7, 8, 9]),
(9, [7, 8]),
(12, [7, 8, 11]),
(8, [7, 11]),
(14, [7, 11, 13]),
(7, [11, 13]),
(21, [11, 13, 15, 16, 17, 18, 19, 20]),
(15, [11, 13, 16, 17, 18, 19, 20]),
(18, [11, 13, 16, 17, 19, 20]),
(16, [11, 13, 17, 19, 20]),
(20, [11, 13, 17, 19]),
(22, [11, 13, 17, 19]),
(11, [13, 17, 19]),
(33, [13, 17, 19, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]),
(24, [13, 17, 19, 23, 25, 26, 27, 28, 29, 30, 31, 32])]
EKG(7):
[(<next>, [<state>]), ...]
[(1, []),
(7, []),
(14, [2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13]),
(2, [3, 4, 5, 6, 8, 9, 10, 11, 12, 13]),
(4, [3, 5, 6, 8, 9, 10, 11, 12, 13]),
(6, [3, 5, 8, 9, 10, 11, 12, 13]),
(3, [5, 8, 9, 10, 11, 12, 13]),
(9, [5, 8, 10, 11, 12, 13]),
(12, [5, 8, 10, 11, 13]),
(8, [5, 10, 11, 13]),
(10, [5, 11, 13]),
(5, [11, 13]),
(15, [11, 13]),
(18, [11, 13, 16, 17]),
(16, [11, 13, 17]),
(20, [11, 13, 17, 19]),
(22, [11, 13, 17, 19, 21]),
(11, [13, 17, 19, 21]),
(33, [13, 17, 19, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]),
(21, [13, 17, 19, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]),
(24, [13, 17, 19, 23, 25, 26, 27, 28, 29, 30, 31, 32])]
REXX
/*REXX program can generate and display several EKG sequences (with various starts).*/
parse arg nums start /*obtain optional arguments from the CL*/
if nums=='' | nums=="," then nums= 50 /*Not specified? Then use the default.*/
if start= '' | start= "," then start=2 5 7 9 10 /* " " " " " " */
do s=1 for words(start); $= /*step through the specified STARTs. */
second= word(start, s); say /*obtain the second integer in the seq.*/
do j=1 for nums
if j<3 then do; #=1; if j==2 then #=second; end /*handle 1st & 2nd number*/
else #= ekg(#)
$= $ right(#, max(2, length(#) ) ) /*append the EKG integer to the $ list.*/
end /*j*/ /* [↑] the RIGHT BIF aligns the numbers*/
say '(start' right(second, max(2, length(second) ) )"):"$ /*display EKG seq.*/
end /*s*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
add_: do while z//j == 0; z=z%j; _=_ j; w=w+1; end; return strip(_)
/*──────────────────────────────────────────────────────────────────────────────────────*/
ekg: procedure expose $; parse arg x 1 z,,_
w=0 /*W: number of factors.*/
do k=1 to 11 by 2; j=k; if j==1 then j=2 /*divide by low primes. */
if j==9 then iterate; call add_ /*skip ÷ 9; add to list.*/
end /*k*/
/*↓ skips multiples of 3*/
do y=0 by 2; j= j + 2 + y//4 /*increment J by 2 or 4.*/
parse var j '' -1 r; if r==5 then iterate /*divisible by five ? */
if j*j>x | j>z then leave /*passed the sqrt(x) ? */
_= add_() /*add a factor to list. */
end /*y*/
j=z; if z\==1 then _= add_() /*Z¬=1? Then add──►list.*/
if _='' then _=x /*Null? Then use prime. */
do j=3; done=1
do k=1 for w
if j // word(_, k)==0 then do; done=0; leave; end
end /*k*/
if done then iterate
if wordpos(j, $)==0 then return j /*return an EKG integer.*/
end /*j*/
{{out|output|text= when using the default inputs:}}
(start 2): 1 2 4 6 3 9 12 8 10 5 15 18 14 7 21 24 16 20 22 11 33 27 30 25 35 28 26 13 39 36 32 34 17 51 42 38 19 57 45 40 44 46 23 69 48 50 52 54 56 49
(start 5): 1 5 10 4 6 3 9 12 8 14 7 21 15 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32 34 17 51 36 38 19 57 42 40 44 46 23 69 45 48 50 52 54 56 49 63
(start 7): 1 7 14 4 6 3 9 12 8 10 5 15 18 16 20 22 11 33 21 24 26 13 39 27 30 25 35 28 32 34 17 51 36 38 19 57 42 40 44 46 23 69 45 48 50 52 54 56 49 63
(start 9): 1 9 3 6 4 8 10 5 15 12 14 7 21 18 16 20 22 11 33 24 26 13 39 27 30 25 35 28 32 34 17 51 36 38 19 57 42 40 44 46 23 69 45 48 50 52 54 56 49 63
(start 10): 1 10 4 6 3 9 12 8 14 7 21 15 5 20 16 18 22 11 33 24 26 13 39 27 30 25 35 28 32 34 17 51 36 38 19 57 42 40 44 46 23 69 45 48 50 52 54 56 49 63
```
## Sidef
{{trans|Perl 6}}
```ruby
class Seq(terms, callback) {
method next {
terms += callback(terms)
}
method nth(n) {
while (terms.len < n) {
self.next
}
terms[n-1]
}
method first(n) {
while (terms.len < n) {
self.next
}
terms.first(n)
}
}
func next_EKG (s) {
2..Inf -> first {|k|
!(s.contains(k) || s[-1].is_coprime(k))
}
}
func EKG (start) {
Seq([1, start], next_EKG)
}
func converge_at(ints) {
var ekgs = ints.map(EKG)
2..Inf -> first {|k|
(ekgs.map { .nth(k) }.uniq.len == 1) &&
(ekgs.map { .first(k).sort }.uniq.len == 1)
}
}
for k in [2, 5, 7, 9, 10] {
say "EKG(#{k}) = #{EKG(k).first(10)}"
}
for arr in [[5,7], [2, 5, 7, 9, 10]] {
var c = converge_at(arr)
say "EKGs of #{arr} converge at term #{c}"
}
```
{{out}}
```txt
EKG(2) = [1, 2, 4, 6, 3, 9, 12, 8, 10, 5]
EKG(5) = [1, 5, 10, 2, 4, 6, 3, 9, 12, 8]
EKG(7) = [1, 7, 14, 2, 4, 6, 3, 9, 12, 8]
EKG(9) = [1, 9, 3, 6, 2, 4, 8, 10, 5, 15]
EKG(10) = [1, 10, 2, 4, 6, 3, 9, 12, 8, 14]
EKGs of [5, 7] converge at term 21
EKGs of [2, 5, 7, 9, 10] converge at term 45
```
## zkl
Using gcd hint from Go.
```zkl
fcn ekgW(N){ // --> iterator
Walker.tweak(fcn(rp,buf,w){
foreach n in (w){
if(rp.value.gcd(n)>1)
{ rp.set(n); w.push(buf.xplode()); buf.clear(); return(n); }
buf.append(n); // save small numbers not used yet
}
}.fp(Ref(N),List(),Walker.chain([2..N-1],[N+1..]))).push(1,N)
}
```
```zkl
foreach n in (T(2,5,7,9,10)){ println("EKG(%2d): %s".fmt(n,ekgW(n).walk(10).concat(","))) }
```
{{out}}
```txt
EKG( 2): 1,2,4,6,3,9,12,8,10,5
EKG( 5): 1,5,10,2,4,6,3,9,12,8
EKG( 7): 1,7,14,2,4,6,3,9,12,8
EKG( 9): 1,9,3,6,2,4,8,10,5,15
EKG(10): 1,10,2,4,6,3,9,12,8,14
```
```zkl
fcn convergeAt(n1,n2,etc){ ns:=vm.arglist;
ekgWs:=ns.apply(ekgW); ekgWs.apply2("next"); // pop initial 1
ekgNs:=List()*vm.numArgs; // ( (ekg(n1)), (ekg(n2)) ...)
do(1_000){ // find convergence in this many terms or bail
ekgN:=ekgWs.apply("next"); // (ekg(n1)[n],ekg(n2)[n] ...)
ekgNs.zipWith(fcn(ns,n){ ns.merge(n) },ekgN); // keep terms sorted
// are all ekg[n]s == and both sequences have same terms?
if(not ekgN.filter1('!=(ekgN[0])) and not ekgNs.filter1('!=(ekgNs[0])) ){
println("EKG(", ns.concat(","), ") converge at term ",ekgNs[0].len() + 1);
return();
}
}
println(ns.concat(",")," don't converge");
}
convergeAt(5,7);
convergeAt(2,5,7,9,10);
```
{{out}}
```txt
EKG(5,7) converge at term 21
EKG(2,5,7,9,10) converge at term 45
```