bibkey: bala2026a295462
authors: Peter Bala
year: 2026
title: "OEIS A295462: binary-spacing parity conjecture"
doi: null
url: https://oeis.org/A295462
claim: "Conjecture: a(n) is odd iff the 1 bits in the binary expansion of n are separated by at least three 0's, that is, iff n belongs to A048718."
strata_touched:
  - D5/S3/Arith/FactorialRatio/BalaA295462Parity
license: citation-only
triage: anchor

A295462 binary-spacing parity

Verified locator

Source: https://oeis.org/A295462

Native record: https://oeis.org/A295462/internal, record #19. The conjecture occurs in the unpaginated REFERENCES field (%D). The related three-page note is Peter Bala, “A295462(n) is divisible by 6n+1 and 10n+1”, https://oeis.org/A295462/a295462.pdf. The target has no arXiv identifier.

Source statements

OEIS A295462, NAME (unpaginated):

a(n) = (15n)!(2n)!/((10n)!(6n)!*n!).

OEIS A295462, REFERENCES (%D, unpaginated):

For n >= 0, both a(n)/(10n+1) and a(n)/(6n+1) are integers.

Conjecture: a(n) is odd iff the 1 bits in the binary expansion of n are separated by at least three 0’s, that is, iff n belongs to A048718.

OEIS A048718, COMMENTS (unpaginated):

Max. 1 one-bit occur in each range of four bits.

The rational quotient uses exact division. Binary position i has weight 2^i; all occupied positions i < j satisfy i+4 ≤ j precisely when three or more zero bits separate successive one bits. Zero has empty support. The natural integer realizing the quotient is required explicitly before testing its oddness.

The linked note proves the two divisibilities independently and contains no parity characterization. The parity proof uses nonnegative floor summands of the binary prime valuation. Separated support makes each summand zero; the first gap of length one, two or three makes one summand equal to one. Completeness of the external literature screen is ASSUMED-UNVERIFIED.