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

Author

Scott R. Shannon, Jan 28 2022

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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}]][[-1, -1]] (* _Michael De Vlieger, Jan 28 2022 *)