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

1,2

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..511

Rémy Sigrist, PARI program for A333403

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.

Example

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

Prog

(PARI) See Links section.

Crossrefs

Cf. A007814, A332941.

Sequence in context: A286260 A226374 A050401 * A271060 A318576 A089276

Adjacent sequences: A333400 A333401 A333402 * A333404 A333405 A333406

Keyword

nonn,base

Author

Rémy Sigrist, Mar 22 2020

Status

approved