A352768 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 both a(n-2)+a(n-1) and a(n-2)*a(n-1).

1, 2, 6, 4, 8, 10, 12, 14, 16, 18, 20, 22, 15, 74, 178, 24, 26, 28, 21, 7, 30, 111, 3, 9, 27, 33, 36, 39, 40, 158, 32, 34, 38, 42, 35, 44, 316, 46, 48, 50, 52, 45, 194, 478, 54, 56, 58, 60, 62, 64, 66, 55, 11, 70, 63, 49, 72, 77, 298, 78, 68, 76, 51, 254, 75, 84, 57, 69, 81, 87, 90, 93, 96, 98

Offset: 1

Author

Scott R. Shannon, Apr 02 2022

Keywords

• nonn

Comments

In the first 250000 terms the lowest number not to appear is 811; it is likely this sequence is a permutation of the positive integers. In this range the fixed points are 1, 2, 4, 35, 6731, 167821. It is possible more exist although this is unknown. In the same range the longest runs of consecutive even terms and odd terms is 29 and 33 respectively. This suggests such runs are likely arbitrarily long.

Examples

			a(3) = 6 as a(1)+a(2)=3, a(1)*a(2)=2, and 6 is the smallest unused number that shares a factor with both 3 and 2.
		

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..2^16, showing records in red, local minima in blue, fixed points in gold, and highlighting primes in green.
• Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.

Crossrefs

Cf. A351001, A349493, A251622.

Programs

Mathematica

nn = 74; c[] = 0; a[1] = c[1] = 1; a[2] = c[1] = 2; u = 1; While[c[u] > 0, u++]; Do[k = u; While[Nand[c[k] == 0, GCD[#1, k] > 1, GCD[#2, k] > 1], k++] & @@ {#1 + #2, #1 #2} & @@ {a[i - 2], a[i - 1]}; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 1, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 02 2022 *)