A333403 - OEIS

%I #21 Mar 29 2020 02:12:22

%S 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,

%T 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,

%U 48,1,8,1,81926,1,8,1,48,1,8,1,68,1,8,1,48,1,8

%N 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).

%C This sequence is a variant of A332941.

%C This sequence is infinite:

%C - suppose that the first n terms are known,

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

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

%C - as there are prime gaps of any size,

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

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

%H Rémy Sigrist, <a href="/A333403/b333403.txt">Table of n, a(n) for n = 1..511</a>

%H Rémy Sigrist, <a href="/A333403/a333403.gp.txt">PARI program for A333403</a>

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

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

%e The values of a(i) XOR ... XOR a(j) for i <= j <= 8 are:

%e   i\j|  1  2  3   4   5   6   7    8

%e   ---+------------------------------

%e     1|  1  9  8  56  57  49  48  116

%e     2|  .  8  9  57  56  48  49  117

%e     3|  .  .  1  49  48  56  57  125

%e     4|  .  .  .  48  49  57  56  124

%e     5|  .  .  .  .    1   9   8   76

%e     6|  .  .  .  .   .    8   9   77

%e     7|  .  .  .  .   .   .    1   69

%e     8|  .  .  .  .   .   .   .    68

%o (PARI) See Links section.

%Y Cf. A007814, A332941.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Mar 22 2020