A297929 Lexicographically earliest sequence of distinct prime numbers such that for any n > 1, a(n) is at Hamming distance one from some previous term.

2, 3, 7, 5, 11, 13, 19, 17, 23, 29, 31, 37, 43, 41, 47, 53, 59, 61, 67, 71, 79, 83, 101, 97, 103, 107, 109, 113, 131, 139, 137, 151, 149, 157, 163, 167, 179, 181, 199, 197, 193, 211, 227, 229, 241, 263, 269, 271, 293, 317, 313, 281, 283, 347, 331, 353, 359

Offset

1,1

Comments

For any n > 1, a(n) = a(m) XOR 2^k for some m < n and k >= 0 (where XOR denotes the bitwise XOR operator).

This sequence was inspired by A294994.

Let define the binary relation R over prime numbers as follows:

- for any prime numbers p and q, p is R-related to q iff there exists a finite list of prime numbers, say (c(1), ..., c(k)), such that c(1) = p and c(k) = q and A000120(c(i) XOR c(i+1)) = 1 for i = 1..k-1,

- R is a equivalence relation,

- this sequence corresponds to the R-equivalence class of the prime number 2.

Is this sequence infinite?

Will every prime number appear?

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Scatterplot of the first 359854 terms (a(359855) is the first term > 2^23)

Rémy Sigrist, Illustration of the first terms

Rémy Sigrist, PARI program for A297929

Example

See illustration of the first terms in Links section.

Mathematica

With[{nn = 56}, Nest[Function[a, Append[a, SelectFirst[Prime@ Range[3 nn/2], Function[p, And[FreeQ[a, p], AnyTrue[a, Total@ IntegerDigits[BitXor[p, #], 2] == 1 &]]]]]], {2}, nn]] (* Michael De Vlieger, Jan 14 2018 *)

Prog

(PARI) See Links section.

Crossrefs

Cf. A000120, A294994.

Sequence in context: A327093 A171039 A063904 * A221858 A139317 A117928

Adjacent sequences: A297926 A297927 A297928 * A297930 A297931 A297932

Keyword

nonn,base

Author

Rémy Sigrist, Jan 08 2018

Status

approved