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A375485 a(n) is the number of integers k between 0 and n such that n XOR k is a prime number (where XOR denotes the bitwise XOR operator).

%I A375485 #8 Aug 18 2024 09:05:28
%S A375485 0,0,2,2,2,2,4,4,2,2,4,4,4,4,6,6,5,5,7,7,7,7,9,9,7,7,9,9,9,9,11,11,7,
%T A375485 7,9,9,9,9,11,11,9,9,11,11,11,11,13,13,12,12,14,14,14,14,16,16,14,14,
%U A375485 16,16,16,16,18,18,13,13,15,15,15,15,17,17,15,15,17
%N A375485 a(n) is the number of integers k between 0 and n such that n XOR k is a prime number (where XOR denotes the bitwise XOR operator).
%H A375485 Rémy Sigrist, <a href="/A375485/b375485.txt">Table of n, a(n) for n = 0..8192</a>
%H A375485 Rémy Sigrist, <a href="/A375485/a375485.png">Scatterplot of (n, k) such that 0 <= k <= n <= 1024 and n XOR k is prime</a>
%F A375485 Empirically, if 2^k <= n < 2^(k+1) then a(n) + a(A054429(n)) only depends on k.
%e A375485 The first terms, alongside the corresponding k's, are:
%e A375485   n   a(n)  k's
%e A375485   --  ----  -------------------
%e A375485    0     0  None
%e A375485    1     0  None
%e A375485    2     2  0, 1
%e A375485    3     2  0, 1
%e A375485    4     2  1, 3
%e A375485    5     2  0, 2
%e A375485    6     4  1, 3, 4, 5
%e A375485    7     4  0, 2, 4, 5
%e A375485    8     2  3, 5
%e A375485    9     2  2, 4
%e A375485   10     4  1, 7, 8, 9
%e A375485   11     4  0, 6, 8, 9
%e A375485   12     4  1, 7, 9, 11
%e A375485   13     4  0, 6, 8, 10
%e A375485   14     6  3, 5, 9, 11, 12, 13
%e A375485   15     6  2, 4, 8, 10, 12, 13
%o A375485 (PARI) a(n) = sum(k = 0, n, isprime(bitxor(n, k)))
%Y A375485 Cf. A054429, A375486 (OR variant), A375487 (AND variant).
%K A375485 nonn,base,easy
%O A375485 0,3
%A A375485 _Rémy Sigrist_, Aug 17 2024