bibkey: kurkov2019a284005
authors: Mikhail Kurkov; Antti Karttunen; OEIS contributors
year: 2019
title: "OEIS A284005: Kurkov's full lower-bit recurrence"
doi: null
url: https://oeis.org/A284005
claim: "Kurkov's November 10, 2019 conjecture: for n>1, with L=floor(log_2(n)), f(n)=n-2^L and T(n,k)=floor(n/2^k) mod 2, a(n)=2*a(f(n))+Sum_{k=0..L-1} a(f(n)+2^k*(1-T(n,k))), for A284005 with a(0)=1, a(1)=2 and a(n)=(1+popcount(n))*a(floor(n/2)) for n>1."
strata_touched:
  - D5/S1/Digit/KurkovLowerBitRecurrence
license: CC-BY-SA-4.0
triage: anchor

OEIS A284005: Kurkov’s lower-bit recurrence

Verified locator

Source: https://oeis.org/A284005, FORMULA, Mikhail Kurkov’s November 10, 2019 conjecture. Entry history: https://oeis.org/history?seq=A284005. The entry is authored by Antti Karttunen; this conjecture is credited to Mikhail Kurkov. OEIS source content and its adapted statement here are attributed to these authors and the OEIS contributors under CC-BY-SA-4.0: https://creativecommons.org/licenses/by-sa/4.0/.

The NAME states, verbatim:

a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)).

The entry’s initial data supplies a(1)=2. A000120(n) is the number of ones in the binary expansion of n. The divisor-count representation in the NAME quotation is not a premise or conclusion of the native result.

Exact question and native result

In normalized notation, for every integer n>1 set L=floor(log_2(n)), f(n)=n-2^L and T(n,k)=floor(n/2^k) mod 2. The conjecture is

a(n) = 2*a(f(n)) + Sum_{k=0..L-1} a(f(n)+2^k*(1-T(n,k))).

Every k from zero through L-1 is included. An existing one bit has T(n,k)=1 and contributes a(f(n)); such repeated contributions remain in the sum. The domain includes n=2 and all powers of two. It requires no value of log_2(0) and asserts no lower-bit identity at n=0 or n=1.

The repository’s native result uses the imported A284005 function and binary digit sum, and contains the initial values, the defining recurrence for every natural n>1, and the complete identity above. Natural-number quantification represents exactly the source’s integer domain n>1. The source supplies the question, not the repository proof. The sequence supplier’s April 24, 2023 two-parameter binomial theorem is a distinct identity and is not used as a substitute for this question.

Literature scope

The bounded original-entry and history reading, author attribution, repository issue/comment reading, and available resolution searches identified no exact prior proof in the inspected scope. This establishes neither worldwide openness nor priority. OpenAlex rate limiting and unavailable StackExchange advanced-search responses leave those surfaces unverified. GitHub code-search authorization was unavailable, so no complete content-level anonymous-equivalent theorem screen is claimed. No DOI or prior proof is attributed to this conjecture.

The OEIS text endpoint reading on 2026-10-28 independently confirms the November 10, 2019 FORMULA, its author, the full summation range, and the initial data. That line still carries the label “Conjecture”. The exact question is recorded in Problems/oeis-a284005-kurkov-lower-bit-recurrence.md; the native result is D5/S1/Digit/KurkovLowerBitRecurrence.result. The label is a source observation, not evidence that no other proof exists.