A337013 -id:A337013 - cp's OEIS frontend

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

Rémy Sigrist, Nov 21 2020

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.

			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

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

See A337013 for the corresponding prime numbers.

See A308334 for similar sequences.

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)))

A352723 a(n) = A352722(n) AND A352722(n+1) (where AND denotes the bitwise AND operator).

Original entry on oeis.org

2, 2, 2, 2, 5, 5, 13, 11, 2, 2, 2, 17, 17, 17, 17, 19, 2, 2, 2, 2, 2, 2, 2, 2, 3, 29, 5, 37, 37, 41, 41, 43, 37, 53, 53, 61, 59, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 17, 17, 17, 17, 5, 5, 5, 5, 73, 73, 71, 83, 89, 89, 73, 73, 67, 97, 97, 101, 101, 109

Offset: 1

Rémy Sigrist, Mar 30 2022

All terms are prime.

			a(5) = A352722(5) AND A352722(6) = 7 AND 5 = 5.

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

Cf. A308340, A337013, A352722.

s=0; v=2; for (n=1, 74, s+=2^v; for (w=1, oo, if (!bittest(s, w) && isprime(p=bitand(v, w)), v=w; print1(p", "); break)))

cp's OEIS Frontend

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).

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