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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). 2
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 (list; graph; refs; listen; history; text; internal format)
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: A165501 A274614 A340859 * A340858 A309364 A062825

Adjacent sequences:  A336814 A336815 A336816 * A336818 A336819 A336820

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Nov 21 2020

STATUS

approved

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Last modified May 9 18:53 EDT 2021. Contains 343744 sequences. (Running on oeis4.)