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

Author

Rémy Sigrist, Jan 08 2018

Keywords

• nonn
• base

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?

Examples

			See illustration of the first terms in Links section.
		

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

Crossrefs

Cf. A000120, A294994.

Programs

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 *)

PARI

See Links section.