A333403 Lexicographically earliest sequence of positive integers such that for any m and n with m <= n, a(m) XOR ... XOR a(n) is neither null nor prime (where XOR denotes the bitwise XOR operator).

1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 1158, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 4752, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 1158, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8, 1, 81926, 1, 8, 1, 48, 1, 8, 1, 68, 1, 8, 1, 48, 1, 8

Offset: 1

Author

Rémy Sigrist, Mar 22 2020

Keywords

• nonn
• base

Comments

This sequence is a variant of A332941.

This sequence is infinite:

- suppose that the first n terms are known,

- let M = max_{k <= n} a(k) XOR ... XOR a(n),

- let k be such that M < 2^k,

- as there are prime gaps of any size,

we can choose an interval of the form [m*2^k..(m+1)*2^k] without prime numbers,

- hence a(n+1) <= m*2^k, QED.

Examples

			The values of a(i) XOR ... XOR a(j) for i <= j <= 8 are:
  i\j|  1  2  3   4   5   6   7    8
  ---+------------------------------
    1|  1  9  8  56  57  49  48  116
    2|  .  8  9  57  56  48  49  117
    3|  .  .  1  49  48  56  57  125
    4|  .  .  .  48  49  57  56  124
    5|  .  .  .  .    1   9   8   76
    6|  .  .  .  .   .    8   9   77
    7|  .  .  .  .   .   .    1   69
    8|  .  .  .  .   .   .   .    68
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..511
• Rémy Sigrist, PARI program for A333403

Crossrefs

Cf. A007814, A332941.

Programs

PARI

See Links section.

Formula

a(m) = a(n) iff A007814(n) = A007814(m).

a(n) = a(2^k-n) for any k >= 0 and n = 1..2^k-1.