A251622 a(1) = 1; a(2) = 2; for n > 2, a(n) = smallest number not already used which shares a factor with a(n-1) or a(n-2).

1, 2, 4, 6, 3, 8, 9, 10, 5, 12, 14, 7, 16, 18, 15, 20, 21, 22, 11, 24, 26, 13, 28, 30, 25, 27, 33, 36, 32, 34, 17, 38, 19, 40, 35, 42, 39, 44, 45, 46, 23, 48, 50, 51, 52, 54, 56, 49, 58, 29, 60, 55, 57, 63, 66, 62, 31, 64, 68, 70, 65, 72, 69, 74, 37, 76, 78, 75, 80

Offset: 1

Author

Franklin T. Adams-Watters, Dec 06 2014

Keywords

• nonn

Comments

This appears certainly to be a permutation of the positive integers.

I believe the arguments we used to show that the EKG sequence (A064413) is a permutation of the natural numbers apply here with almost no change. - N. J. A. Sloane, Jan 02 2015

For n > 2: gcd(a(n),a(n-1)*a(n-2)) > 1. - Reinhard Zumkeller, Apr 05 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
• J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG Sequence

Crossrefs

Cf. A064413, A089088, A098550.

Cf. A256628 (conjectured inverse).

Programs

Haskell

a251622 n = a251622_list !! (n-1)
a251622_list = 1 : 2 : f 1 2 [3..] where
   f u v xs = g xs where
     g (w:ws) = if gcd w u > 1 || gcd w v > 1
                   then w : f v w (delete w xs) else g ws
-- Reinhard Zumkeller, Apr 05 2015

Mathematica

a[1] = 1; a[2] = 2;
a[n_] := a[n] = For[k = 1, True, k++, If[FreeQ[Array[a, n-1], k], If[!CoprimeQ[k, a[n-1]] || !CoprimeQ[k, a[n-2]], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Sep 05 2018 *)

PARI

invecn(v,k,x)=for(i=1,k,if(v[i]==x,return(i)));0
alist(n)=local(v=vector(n),x);v[1]=1;v[2]=2;for(k=3,n,x=3;while(invecn(v,k-1,x)||(gcd(v[k-1],x)==1&&gcd(v[k-2],x)==1),x++);v[k]=x);v