A375486 -id:A375486 - cp's OEIS frontend

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).

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, 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, 16, 16, 16, 16, 18, 18, 13, 13, 15, 15, 15, 15, 17, 17, 15, 15, 17

Offset: 0

Rémy Sigrist, Aug 17 2024

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  -------------------
   0     0  None
   1     0  None
   2     2  0, 1
   3     2  0, 1
   4     2  1, 3
   5     2  0, 2
   6     4  1, 3, 4, 5
   7     4  0, 2, 4, 5
   8     2  3, 5
   9     2  2, 4
  10     4  1, 7, 8, 9
  11     4  0, 6, 8, 9
  12     4  1, 7, 9, 11
  13     4  0, 6, 8, 10
  14     6  3, 5, 9, 11, 12, 13
  15     6  2, 4, 8, 10, 12, 13

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of (n, k) such that 0 <= k <= n <= 1024 and n XOR k is prime

Cf. A054429, A375486 (OR variant), A375487 (AND variant).

a(n) = sum(k = 0, n, isprime(bitxor(n, k)))

Empirically, if 2^k <= n < 2^(k+1) then a(n) + a(A054429(n)) only depends on k.

A375487 a(n) is the number of integers k between 0 and n such that n AND k is a prime number (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 4, 0, 0, 4, 5, 0, 3, 2, 6, 0, 1, 8, 10, 0, 6, 4, 11, 0, 4, 4, 10, 0, 7, 2, 11, 0, 0, 16, 16, 0, 9, 8, 17, 0, 1, 8, 14, 0, 12, 4, 16, 0, 8, 8, 16, 0, 13, 4, 17, 0, 8, 4, 15, 0, 15, 2, 18, 0, 0, 32, 33, 0, 16, 16, 34, 0, 1, 16, 27, 0, 18, 8

Offset: 0

Rémy Sigrist, Aug 17 2024

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ------------------
   0     0  None
   1     0  None
   2     1  2
   3     2  2, 3
   4     0  None
   5     1  5
   6     2  2, 3
   7     4  2, 3, 5, 7
   8     0  None
   9     0  None
  10     4  2, 3, 6, 7
  11     5  2, 3, 6, 7, 11
  12     0  None
  13     3  5, 7, 13
  14     2  2, 3
  15     6  2, 3, 5, 7, 11, 13

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of (n, k) such that 0 <= k <= n <= 1024 and n AND k is prime

Cf. A000720, A102210, A117494, A375485 (XOR variant), A375486 (OR variant).

a(n) = sum(k = 0, n, isprime(bitand(n, k)))

a(n) = 0 iff n = 0 or n belongs to A102210.

a(2^k-1) = A000720(2^k-1) for any k > 0.

cp's OEIS Frontend

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).

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