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a(1)=1, a(2)=2, a(3)=3; for n > 3, a(n) is the smallest positive integer which does not occur earlier in the sequence and which is not coprime to a(n-2).
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%I #44 Feb 03 2019 11:42:30

%S 1,2,3,4,6,8,9,10,12,5,14,15,7,18,21,16,24,20,22,25,11,30,33,26,27,13,

%T 36,39,28,42,32,34,38,17,19,51,57,45,48,35,40,49,44,56,46,50,23,52,69,

%U 54,60,58,55,29,65,87,70,63,62,66,31,64,93,68,72,74,75,37,78,111,76,81

%N a(1)=1, a(2)=2, a(3)=3; for n > 3, a(n) is the smallest positive integer which does not occur earlier in the sequence and which is not coprime to a(n-2).

%C Conjecture: this is a permutation of the positive integers, cf. A256618. - _Reinhard Zumkeller_, Apr 05 2015

%C The B-sequence mentioned in the Maple program is not in the OEIS. It is 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, ... - _Alois P. Heinz_, Feb 02 2019

%H N. J. A. Sloane, <a href="/A121217/b121217.txt">Table of n, a(n) for n = 1..20632</a>

%p # From _N. J. A. Sloane_, Apr 04 2015: A121217 gcd(A[n],A[n-2])>1 A=seq, for B see the COMMENTS

%p N:= 60: # to get a(1) to a(n) where a(n+1) is the first term > N

%p B:= Vector(N, datatype=integer[4]):

%p for n from 1 to 3 do A[n]:= n: od:

%p for n from 4 do

%p for k from 4 to N do

%p if B[k] = 0 and igcd(k, A[n-2]) > 1 then

%p A[n]:= k;

%p B[k]:= 1;

%p break

%p fi

%p od:

%p if k > N then break fi

%p od:

%p [seq(A[i], i=1..n-1)];

%t a = Range@ 3; Do[k = 4; While[Or[MemberQ[a, k], CoprimeQ[a[[i - 2]], k]], k++]; AppendTo[a, k], {i, 4, 72}]; a (* _Michael De Vlieger_, Aug 19 2017 *)

%o (Haskell)

%o a121217 n = a121217_list !! (n-1)

%o a121217_list = 1 : 2 : 3 : f 2 3 [4..] where

%o f u v xs = g xs where

%o g (w:ws) = if gcd w u > 1 then w : f v w (delete w xs) else g ws

%o -- _Reinhard Zumkeller_, Apr 05 2015

%Y Cf. A064413, A121216, A251622, A256414 (indices of primes), A256419 (smoothed version).

%Y Cf. A256618 (conjectured inverse).

%K nonn

%O 1,2

%A _Leroy Quet_, Aug 20 2006

%E Extended by _Ray Chandler_, Aug 22 2006