A354087 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and whose binary expansion has a single 1-bit in common with the binary expansion of a(n-1).

1, 3, 6, 2, 10, 8, 12, 4, 14, 18, 15, 20, 5, 25, 35, 21, 9, 24, 16, 22, 11, 33, 27, 48, 26, 13, 52, 32, 34, 30, 36, 28, 7, 42, 49, 77, 56, 38, 19, 133, 57, 69, 46, 66, 39, 65, 45, 50, 40, 54, 68, 44, 70, 58, 72, 60, 74, 64, 76, 80, 55, 88, 96, 51, 78, 81, 102, 130, 62, 132, 63, 129, 43, 86, 104, 82

Offset

1,2

Comments

This sequence is similar to the EKG sequence A064413 with the additional restriction that each term must share a single 1-bit in common with the previous term in their binary expansions. These restrictions lead to numerous terms being significantly 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 both the proceeding and following terms of the primes can be large multiples of the prime.

In the first 100000 terms the fixed points are 1, 3, 30, 38, 350, 1603, 1936, 10176, 11976, 46123, 58471, 89870, although it is likely more exist. In the same range the lowest unseen number is 1019; the sequence is conjectured to be a permutation of the positive integers.

Links

Table of n, a(n) for n=1..76.

Scott R. Shannon, Image of the first 100000 terms for values less than 200000. The green line is y = n.

Example

a(6) = 8 as a(5) = 10, 8 = 1000_2, 10 = 1010_2, and 8 is the smallest unused number that shares a common factor with 10 and has a single 1-bit in common with 10 in their binary expansions. Note that 4 satisfies the first criterion but not the second.

Crossrefs

Cf. A353989, A064413, A353245, A152458, A353712, A352633.

Sequence in context: A257106 A360255 A353989 * A210187 A364365 A210189

Adjacent sequences: A354084 A354085 A354086 * A354088 A354089 A354090

Keyword

nonn,base

Author

Scott R. Shannon, May 17 2022

Status

approved