A353989 a(1) = 1; a(2) = 3; for n > 2, a(n) is the smallest positive number that has not appeared that shares a factor with a(n-1) and whose binary expansion has a 1-bit in common with the binary expansion of a(n-1).

1, 3, 6, 2, 10, 8, 12, 4, 14, 7, 21, 9, 15, 5, 20, 16, 18, 22, 11, 33, 27, 24, 26, 13, 39, 30, 25, 35, 40, 28, 36, 32, 34, 38, 19, 57, 42, 44, 46, 23, 69, 45, 48, 50, 52, 54, 51, 17, 85, 55, 60, 56, 49, 63, 66, 58, 29, 87, 72, 62, 31, 93, 75, 65, 70, 64, 68, 74, 76, 78, 80, 82, 84, 77, 88, 86, 43

Offset: 1

Author

Scott R. Shannon, May 13 2022

Keywords

• nonn

Comments

This sequence is similar to the EKG sequence A064413 with the additional restriction that each term must share at least one 1-bit in common with the previous term in their binary expansions. The majority of terms are concentrated along the same three lines as in A064413 although at least three additional lines appear that contains fewer terms. See the linked image. Unlike A064413 the primes do not occur in their natural order and a prime p can be preceded and followed by multiples of p other than 2p and 3p respectively.

In the first 100000 terms the fixed points are 1, 16, 32, 209, 527, and it is likely no more exist. In the same range the lowest unseen number is 34849; the sequence is conjectured to be a permutation of the positive integers.

See A353245 for the binary AND operation of each pair of terms.

Examples

			a(3) = 6 as a(2) = 3, 6 = 110_2, 3 = 11_2, and 6 is the smallest unused number that shares a common factor with 3 and has a 1-bit in common with 3 in their binary expansions.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Crossrefs

Cf. A064413, A353245, A152458, A353712, A352633.