A357759 Numbers k such that in the binary expansion of k, the Hamming weight of each block differs by at most 2 from every other block of the same length.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75

Offset

1,3

Comments

Leading zeros in binary expansions are ignored.

For any n > 0, there are A274005(n)/2 positive terms with binary length n.

Empirically, if t is a term, then at least one of 2*t or 2*t + 1 is also a term.

If t is a term, then floor(t/2) is also a term.

The complement with respect to the nonnegative integers is 56, 71, 112, 113, 120, 135, 142, 143, 176, 184, 199, ... - Andrew Howroyd, Oct 09 2024

Links

Andrew Howroyd, Table of n, a(n) for n = 1..7658

Rémy Sigrist, PARI program

Index entries for sequences related to binary expansion of n

Example

For k = 12:
- the binary expansion of k is "1100",
- blocks of length 1 have Hamming weight 0 or 1,
- blocks of length 2 have Hamming weight 0, 1 or 2,
- blocks of length 3 have Hamming weight 1 or 2,
- blocks of length 4 have Hamming weight 2,
- so 12 belongs to the sequence.
For k = 56:
- the binary expansion of 44 is "111000",
- blocks of length 3 have Hamming weight 0, 1, 2 or 3,
- so 56 does not belong to the sequence.

Prog

(PARI) \\ See Links section.
(Python)
def ok(n):
    b = bin(n)[2:]
    if "000" in b and "111" in b: return False
    for l in range(4, len(b)-1):
        h = set(b[i:i+l].count("1") for i in range(len(b)-l+1))
        if max(h) - min(h) > 2: return False
    return True
print([k for k in range(69) if ok(k)]) # Michael S. Branicky, Oct 12 2022

Crossrefs

Cf. A274005, A357758.

Sequence in context: A272118 A247163 A376668 * A120951 A272554 A179797

Adjacent sequences: A357756 A357757 A357758 * A357760 A357761 A357762

Keyword

nonn,base

Author

Rémy Sigrist, Oct 12 2022

Extensions

a(69) onwards from Andrew Howroyd, Oct 09 2024

Status

approved