A336817 Lexicographically earliest sequence of distinct positive numbers such that for any n > 0, a(n) XOR a(n+1) is a prime number (where XOR denotes the bitwise XOR operator).

1, 2, 5, 6, 3, 4, 7, 10, 8, 11, 9, 12, 14, 13, 15, 16, 18, 17, 19, 20, 22, 21, 23, 26, 24, 27, 25, 28, 30, 29, 31, 34, 32, 35, 33, 36, 38, 37, 39, 42, 40, 43, 41, 44, 46, 45, 47, 48, 50, 49, 51, 52, 54, 53, 55, 58, 56, 59, 57, 60, 62, 61, 63, 64, 66, 65, 67

Offset

1,2

Comments

By Dirichlet's theorem on arithmetic progressions, we can always extend the sequence: say a(n) < 2^k:

- if a(n) is odd: a(n) and 2^k are coprime and there are infinitely many prime numbers of the form a(n) + m*2^k = a(n) XOR m*2^k, and we can extend the sequence,

- if a(n) is even: a(n)+1 and 2^k are coprime and there are infinitely many prime numbers of the form a(n)+1 + m*2^k = a(n) XOR (1+m*2^k), and we can extend the sequence.

Links

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

Example

The first terms, alongside the corresponding prime numbers, are:
  n   a(n)  a(n) XOR a(n+1)
  --  ----  ---------------
   1     1                3
   2     2                7
   3     5                3
   4     6                5
   5     3                7
   6     4                3
   7     7               13
   8    10                2
   9     8                3
  10    11                2

Prog

(PARI) s=0; v=1; for (n=1, 67, print1 (v ", "); s+=2^v; for (w=1, oo, if (!bittest(s, w) && isprime(bitxor(v, w)), v=w; break)))

Crossrefs

See A337013 for the corresponding prime numbers.

See A308334 for similar sequences.

Sequence in context: A274614 A353771 A340859 * A340858 A309364 A062825

Adjacent sequences: A336814 A336815 A336816 * A336818 A336819 A336820

Keyword

nonn,base

Author

Rémy Sigrist, Nov 21 2020

Status

approved