This is a pretty run of the mill question. I will define a sequence and you golf some code to output a entry given a index.

  • The first item in the sequence is 2.

  • The nth item in the sequence is the smallest positive integer other than n and 1 sharing at least one factor with n (other than 1) that has not already appeared in the list.

Test cases

Here are the first 25 items in the sequence:

1  2
2  4
3  6
4  8
5  10
6  3
7  14
8  12
9  15
10 5
11 22
12 9
13 26
14 7
15 18
16 20
17 34
18 16
19 38
20 24
21 27
22 11
23 46
24 21
25 30

Related (offset by one) OEIS


Jelly, 19 bytes


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  • \$\begingroup\$ I was so happy that I had you out-golfed on the Lyndon word factorization question but then you hit me with this... (in all seriousness though this is a great answer) \$\endgroup\$ – notjagan Jun 20 '17 at 2:50

Python 3, 118 117 bytes

-1 byte thanks to Cameron Aavik!

import math
def f(n,i=3):
 if n<2:return 2
 while 1:
  if math.gcd(n,i)>1>(i in map(f,range(n)))<i!=n:return i

Try it online!

The code is pretty inefficient (it brute-forces a value that doesn't exist in the previous results, and calculates the previous results again on every new value), so it works properly but I wouldn't recommend running it on large numbers.

  • 2
    \$\begingroup\$ Small tip: You can save a newline by making it def f(n,i=3): and removing the i=3 line \$\endgroup\$ – Cameron Aavik Jun 20 '17 at 6:09
  • \$\begingroup\$ 115 bytes. \$\endgroup\$ – Jonathan Frech Sep 25 '17 at 3:29

Haskell, 60 59 bytes


  • -1 byte: @xnor pointed out all(/=x) was shorter than x`notElem`.

f takes an integer and returns an integer.

f n=[x|x<-[2..],gcd x n>1||n<2,all(/=x)$n:map f[1..n-1]]!!0

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This is very exponential time, so TIO times out after 21, while my interpreted GHCi got up to 22 before I stopped it just now. The following 9 bytes longer version memorizing into a list easily goes up into the thousands:

f n=[x|x<-[2..],gcd x n>1||n<2,all(/=x)$n:take(n-1)l]!!0

Try it online!

  • f n uses a list comprehension to generate candidates x, taking the first passing one with !!0.
  • gcd x n>1 checks that x and n have common factors.
  • ||n<2 exempts n==1 from the factor requirement.
  • all(/=x)$n:map f[1..n-1] checks that x is neither n nor a preceding sequence element.
  • \$\begingroup\$ @WheatWizard Hm, probably no difference in that case. Just used to doing it by default. It's one of the few alphanumerical functions that has a fixity defined to fit well that way. \$\endgroup\$ – Ørjan Johansen Jun 20 '17 at 3:41
  • 1
    \$\begingroup\$ all(/=x)$ is 1 shorter there \$\endgroup\$ – xnor Jun 20 '17 at 5:34

No built-in for GCD in C#, so...

C# (.NET Core), 197 196 194 bytes

n=>{if(n<2)return 2;var p=new int[n-1];int i=0,a,b;for(;i<n-1;)p[i]=f(++i);for(i=2;;i++)if(n!=i){for(a=n,b=i;a*b>0;)if(a>b)a%=b;else b%=a;if(b!=1&a!=1&!System.Array.Exists(p,e=>e==i))return i;}}

Try it online!

Once again, refrain from using this code to calculate numbers in the sequence for n>30...

  • -1 byte by changing the GCD while loop for a for loop.
  • -2 bytes thanks to Kevin Cruijssen! Nice one!
  • 1
    \$\begingroup\$ a>0&b>0 can be golfed to a*b>0 \$\endgroup\$ – Kevin Cruijssen Sep 25 '17 at 7:21

APL (Dyalog), 46 bytes


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