A353006 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with (a(n-1) + a(n-2))*|a(n-1) - a(n-2)|.

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 12, 14, 13, 15, 16, 31, 18, 21, 24, 20, 11, 27, 22, 25, 30, 33, 28, 35, 36, 71, 40, 37, 39, 26, 45, 38, 42, 32, 34, 44, 46, 48, 47, 50, 51, 101, 52, 17, 23, 54, 49, 55, 56, 57, 113, 58, 60, 59, 63, 61, 62, 41, 66, 65, 131, 64, 67, 69, 68, 137, 70, 72, 74

Offset: 1

Author

Scott R. Shannon, Apr 16 2022

Keywords

• nonn

Comments

The sequences is conjectured to be a permutation of the positive integers. In the first 500000 terms there are 637 fixed points: 1, 2, 3, 6, 9, ..., 180611, 189383, 298097. As the terms slowly move away from the line a(n) = n as n increases it is likely no more exist.

Examples

			a(4) = 5 as (a(3)+a(2))*|a(3)-a(2)| = (3+2)*|3-2| = 5, and 5 is the smallest unused number that shares a factor with 5.
a(5) = 4 as (a(4)+a(3))*|a(4)-a(3)| = (5+3)*|5-3| = 16, and 4 is the smallest unused number that shares a factor with 16.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Annotated log-log scatterplot of a(n), n - 1..2^14, showing records in red, local minima in blue, fixed points highlighted in amber, and primes in green.
• Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Crossrefs

Cf. A337136, A352867, A352790, A350927.

Programs

Mathematica

nn = 74, c[] = 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; u = 3; Do[m = (#1 + #2) Abs[(#1 - #2)] & @@ {a[i - 1], a[i - 2]}; k = u; While[Nand[c[k] == 0, ! CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 17 2022 *)