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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

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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.