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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.
1
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, Scatterplot of the first 359854 terms (a(359855) is the first term > 2^23)
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
Sequence in context: A327093 A171039 A063904 * A221858 A139317 A117928
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Jan 08 2018
STATUS
approved