bibkey: karttunen2017a284005
authors: Antti Karttunen; Mikhail Kurkov
year: 2023
title: "OEIS A284005, product of (1 + binary weight) over the binary shifts of n, with Kurkov's binomial conjecture"
doi: null
url: https://oeis.org/A284005
claim: "A284005 %N: a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)). A284005 %F: Conjecture: a(2^m*(2n+1)) = Sum_{k=0..m+1} binomial(m+1, k)*a(2^k*n) for m >= 0, n >= 0 with a(0) = 1. - _Mikhail Kurkov_, Apr 24 2023"
strata_touched:
  - D5/S1/Digit/KurkovWeightProductBinomial
license: citation-only
triage: anchor

OEIS A284005: binary-weight products

A000120 is the number of ones in the binary expansion. Starting with a(0)=1, A284005 multiplies one plus this weight along the positive binary shifts. The NAME recurrence is stated for n>1; the sequence data supplies a(1)=2, agreeing with its extension to all positive indices. The initial terms are 1,2,4,6,8,12,18,24,16,24,36,48,54,72,96,120.

The binary weight is the frozen D5/S1/Digit/DyadicRowPolynomialRecurrence.wt, reused by import.

Verified locator

The locator lines below were checked against https://oeis.org/A284005 on 2026-09-15.

• URL: https://oeis.org/A284005
• %N (verbatim): a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)).
• Settled %F (verbatim): Conjecture: a(2^m*(2n+1)) = Sum_{k=0..m+1} binomial(m+1, k)a(2^kn) for m >= 0, n >= 0 with a(0) = 1. - Mikhail Kurkov, Apr 24 2023
• %A (verbatim): Antti Karttunen, Mar 18 2017

No literature proof was found in the searched surfaces (OEIS text, OpenAlex, Math.SE, formal-conjectures); this is a bounded search result, not an exhaustive absence or priority claim.

Scope

Only the quoted April 24, 2023 binomial identity is asserted. The 2019 bit-flip recursion, the other 2023 conjecture about the mod-2 binomial transform of A329369, and the representation A000005(A283477(n)) are not claimed. The latter is retained above only as part of the verbatim NAME quotation. Historical openness outside the searched surfaces is unverified.