A384956 Binary XOR of number of 1-bits in the binary representation of n and number of 0-bits in the binary representation of n, a(0) = 1.

1, 1, 0, 2, 3, 3, 3, 3, 2, 0, 0, 2, 0, 2, 2, 4, 5, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 5, 5, 4, 6, 6, 0, 6, 0, 0, 6, 6, 0, 0, 6, 0, 6, 6, 4, 6, 0, 0, 6, 0, 6, 6, 4, 0, 6, 6, 4, 6, 4, 4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Author

Frederik P.J. Vandecasteele, Jun 13 2025

Keywords

• base
• easy
• look
• nonn

Comments

n=0 is taken to be a single 0 bit, and all other n are taken without leading 0 bits.

When the length of the binary representation of n is 2^k-1, then a(n) is 2^k-1.

Examples

			179 (10110011_2) has 5 (101_2) 1-bits and 3 (011_2) 0-bits. 101_2 XOR 011_2 = 110_2 = 6. a(179) = 6.
		

Links

• Karl-Heinz Hofmann, Log scatter plot up to 2^22

Crossrefs

Cf. A000120, A023416, A070939, A328568.

Cf. A003987 (XOR).

Programs

Mathematica

a[n_] := BitXor @@ DigitCount[n, 2]; Array[a, 100, 0] (* Amiram Eldar, Jun 13 2025 *)

Python

def A384956(n):
    if n == 0 : return 1
    return (n.bit_length() - (Ham:=n.bit_count())) ^ Ham # Karl-Heinz Hofmann, Jun 14 2025

Formula

a(n) = A000120(n) XOR A023416(n).