A369281 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, A091255(a(n), a(n+1)) = 1.

1, 2, 3, 4, 5, 7, 6, 11, 8, 9, 13, 10, 19, 12, 21, 15, 14, 17, 16, 23, 22, 25, 18, 31, 20, 35, 24, 37, 26, 27, 32, 29, 28, 33, 38, 39, 41, 30, 47, 34, 49, 40, 55, 36, 59, 42, 43, 44, 45, 50, 51, 52, 53, 56, 57, 61, 46, 67, 48, 69, 54, 73, 58, 79, 60, 81, 62

Offset: 1

Author

Rémy Sigrist, Jan 18 2024

Keywords

• nonn
• base

Comments

In other words, the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive terms are coprime.

As the polynomials over GF(2) whose coefficients are encoded in the binary expansions of two consecutive integers are not necessarily coprime (see A369317-A369318), the present sequence does not equal the identity map.

This sequence is a permutation of the positive integers with inverse A369282:

- we can always extend the sequence with some term of A014580 not yet in the sequence, hence the sequence is infinite, and all terms of A014580 appear in the sequence, in ascending order,

- for any k > 0, the first term >= A014580(k) is precisely A014580(k),

- if a(n) = A014580(k) for some n and the least value not among the first n terms, say u, is less than A014580(k), then a(n+1) = u,

- and eventually every integer will appear in the sequence.

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Scatterplot of the first 2500 terms.
• Rémy Sigrist, PARI program.
• Index entries for sequences operating on GF(2)[X]-polynomials
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

See A369293 for a similar sequence.

Cf. A014580, A091255, A369282 (inverse), A369317-A369318.

Programs

PARI

See Links section.