A351001 a(0) = 0, a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared which has a common factor with a(n-2) + a(n-1) but does not equal a(n-2) + a(n-1).

0, 1, 2, 6, 4, 5, 3, 10, 26, 8, 12, 14, 13, 9, 11, 15, 16, 62, 18, 20, 19, 21, 22, 86, 24, 25, 7, 28, 30, 29, 118, 27, 35, 31, 32, 33, 39, 34, 146, 36, 38, 37, 40, 42, 41, 166, 23, 45, 17, 44, 122, 46, 48, 47, 50, 194, 52, 51, 206, 514, 54, 56, 55, 57, 49, 53, 58, 60, 59, 63, 61, 64

Offset

0,3

Comments

This is a permutation of the natural numbers. Up to 500000 terms the fixed points are 0, 1, 2, 4, 5, 15, 16, 18, 21, 22, 24, 25, 29, and it is likely no more exist.

Links

Table of n, a(n) for n=0..71.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Example

a(3) = 6 as a(1)+a(2) = 3, 6 does not equal 3, and gcd(3,6) > 1.
a(4) = 4 as a(2)+a(3) = 8, 4 does not equal 8, and gcd(8,4) > 1.

Mathematica

s = {0, 1, 2}; u = 3; c[_] = 0; Set[{i, j}, s[[-2 ;; -1]]]; Array[Set[c[s[[#]]], #] &, Length[s]]; s~Join~Reap[Do[Set[k, u]; While[Nand[c[k] == 0, GCD[i + j, k] > 1, i + j != k], k++]; Sow[k]; Set[c[k], n]; If[k == u, While[c[u] == 1, u++]]; i = j; j = k, {n, Length[s] + 1, 2^10}], n]][[-1, -1]] (* Michael De Vlieger, Jan 28 2022 *)

Crossrefs

Cf. A337136, A064413, A336957, A098550, A089088, A251622.

Sequence in context: A348669 A195730 A222236 * A334171 A175386 A021382

Adjacent sequences: A350998 A350999 A351000 * A351002 A351003 A351004

Keyword

nonn

Author

Scott R. Shannon, Jan 28 2022

Status

approved