A335587 a(n) is the sum of the numbers k such that 0 <= k <= n and n AND k = 0 (where AND denotes the bitwise AND operator).

0, 0, 1, 0, 6, 2, 1, 0, 28, 12, 10, 4, 6, 2, 1, 0, 120, 56, 52, 24, 44, 20, 18, 8, 28, 12, 10, 4, 6, 2, 1, 0, 496, 240, 232, 112, 216, 104, 100, 48, 184, 88, 84, 40, 76, 36, 34, 16, 120, 56, 52, 24, 44, 20, 18, 8, 28, 12, 10, 4, 6, 2, 1, 0, 2016, 992, 976, 480

Offset: 0

Author

Rémy Sigrist, Apr 21 2021

Keywords

• nonn
• base

Comments

All terms can be written as m * 2^A000120(m) for some m >= 0.

Examples

			For n = 4:
- 4 AND 0 = 0,
- 4 AND 1 = 0,
- 4 AND 2 = 0,
- 4 AND 3 = 0,
- 4 AND 4 = 4,
- so a(4) = 0 + 1 + 2 + 3 = 6.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8191
• Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n < 2^16 (where the color is function of A080791(n))
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A004198 (bitwise AND), A006516, A035327, A080100, A080791.

Programs

PARI

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

PARI

a(n) = my (w=#binary(n)); ( (2^w-1-n) * 2^(w-hammingweight(n)) ) \ 2

Formula

a(n) = A035327(n) * A080100(n) / 2 for any n > 0.

a(2*n+1) = 2*a(n).

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

a(2^k) = A006516(k) for any k >= 0.