A361314 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number which has not appeared that shares a factor with a(n-2) + a(n-1) while the sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

1, 2, 3, 5, 4, 6, 8, 7, 9, 10, 19, 29, 12, 41, 53, 14, 67, 15, 16, 31, 47, 13, 20, 18, 22, 24, 26, 25, 17, 27, 28, 11, 21, 36, 30, 32, 38, 34, 39, 73, 35, 33, 42, 45, 48, 51, 44, 40, 46, 43, 89, 50, 139, 49, 52, 101, 54, 55, 109, 56, 57, 113, 58, 60, 59, 63, 61, 62, 66, 64, 65, 69, 68, 137, 70

Offset: 1

Author

Scott R. Shannon, Mar 08 2023

Keywords

• nonn

Comments

In the first 100000 terms the fixed points are 1, 2, 3, 6, 9, 10, 39, 91, 112; it is likely no more exist. The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(23) = 20 as a(21) + a(22) = 47 + 13 = 60, and 20 is the smallest unused number that shares a factor with 60 while the sum a(22) + 20 = 13 + 20 = 33 is distinct from all previous sums. Note that 18 is unused and shares a factor with 60 but the sum a(22) + 18 = 13 + 18 = 31 is the same as a(18) + a(19) = 15 + 16 = 31. This is the first term that differs from A337136.
		

Links

• Scott R. Shannon, Image for n = 1..100000. The green line is a(n) = n.

Crossrefs

Cf. A337136, A359799, A354755, A354727, A354687, A354753, A353989, A354087, A352763.