A337136 - OEIS

A337136

a(1) = 1, a(2) = 2; for n > 1, a(n) = smallest positive number which has not yet appeared which has a common factor with a(n-2) + a(n-1).

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 29, 12, 41, 53, 14, 67, 15, 16, 31, 47, 13, 18, 62, 20, 22, 21, 43, 24, 134, 26, 25, 17, 27, 11, 28, 30, 32, 34, 33, 201, 36, 39, 35, 37, 38, 40, 42, 44, 46, 45, 49, 48, 97, 50, 51, 101, 52, 54, 56, 55, 57, 58, 23, 60, 83, 65

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: This is a permutation of the natural numbers. Up to 1 million terms the only fixed points are 1,2,3,6,9,10, and it is likely that there are no others.

LINKS

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log log scatterplot of a(n), n = 1..1024, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, highlighting in light blue those numbers in the last mentioned category whose prime power exponents all exceed 1.

Scott R. Shannon, Image of the first 1000000 terms. The green line is y=x.

EXAMPLE

a(5) = 4 as a(3) + a(4) = 3 + 5 = 8, and 4 is the smallest number that shares a common factor with 8 that has not yet appeared.

a(11) = 19 as a(9) + a(10) = 9 + 10 = 19, and as 19 is prime, a(11) must be the smallest multiple of 19 that has not yet appeared, being 19 in this case.

MATHEMATICA

nn = 120;

c[_] := False;

Array[Set[{a[#], c[#]}, {#, True}] &, 2]; Set[{i, j, u},

Range[3]]; s = i + j;

Do[k = u;

While[Or[c[k], CoprimeQ[s, k]], k++];

Set[{a[n], c[k], i, j, s}, {k, True, j, k, j + k}];

If[k == u, While[c[u], u++]], {n, 3, nn}];

Array[a, nn] (* Michael De Vlieger, Oct 26 2023 *)

PROG

(PARI) lista(nn) = v=[1, 2]; for(n=3, nn, t=1; while(prod(X=1, n-1, v[X]-t)==0 || gcd(v[n-2]+v[n-1], t)==1, t++); v=concat(v, t); ); v; \\ Yifan Xie, May 18 2023

CROSSREFS

Cf. A064413, A336957, A098550, A089088, A251622.