A352763 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) and whose binary expansion has no 1-bit in common with the binary expansion of a(n-1).

1, 2, 4, 8, 6, 9, 18, 12, 3, 24, 32, 10, 5, 40, 16, 14, 48, 15, 80, 34, 17, 68, 26, 36, 27, 96, 20, 35, 28, 64, 22, 33, 30, 65, 50, 72, 21, 42, 69, 138, 52, 66, 44, 82, 41, 656, 38, 88, 128, 46, 144, 39, 192, 45, 130, 13, 208, 256, 54, 129, 60, 194, 56, 7, 112, 132, 11, 176, 70, 25, 100, 136, 51

Offset: 1

Author

Scott R. Shannon, May 15 2022

Keywords

• nonn

Comments

This sequence is similar to the EKG sequence A064413 with the additional restriction that no term can have a 1-bit in common with the previous term in their binary expansions. These restrictions lead to numerous terms being much larger than their preceding term, while the smaller terms overall show similar behavior to A109812. See the linked image. Unlike A064413 the primes do not occur in their natural order and the term following a prime can be a very large multiple of the prime.

In the first 50000 terms the fixed points are 1, 2, 105, 135, 225, 2157, 3972, 7009, 8531, although it is likely more exist. In the same range the lowest unseen number is 383; the sequence is conjectured to be a permutation of the positive integers.

Examples

			a(5) = 6 as a(4) = 8, 6 = 110_2, 8 = 1000_2, and 6 is the smallest unused number that shares a common factor with 8 but has no 1-bit in common with 8 in their binary expansions.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 10000 terms for values below 15000. The green line is y = n. The maximum value in this range is a(8045) = 959024.

Crossrefs

Cf. A353989, A109812, A064413, A152458, A353712, A352633.