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

0,5

Comments

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

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

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.

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A120002 A296432 A171542 * A320302 A284761 A021165

Adjacent sequences: A335584 A335585 A335586 * A335588 A335589 A335590

Keyword

nonn,base

Author

Rémy Sigrist, Apr 21 2021

Status

approved