bibkey: schulte2018a001222 authors: Werner Schulte year: 2018 title: “OEIS A001222, the Ω/ω binomial convolution conjecture (x+y)^Ω(n) = Σ_{d|n} x^Ω(d) (x+y)^{Ω(n/d)−ω(n/d)} y^{ω(n/d)}” doi: null url: https://oeis.org/A001222 claim: “Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - Werner Schulte, Feb 10 2018” strata_touched:

• D5/S3/Arith/SchulteOmegaBinomialConvolution license: citation-only triage: anchor

OEIS A001222

Schulte’s conjecture identifies the binomial power of the total prime-factor count with a divisor convolution involving the total and distinct counts. The claim is the quoted Conjecture sentence for all complex x, y and positive natural n. Natural exponentiation gives 0^0 = 1, including at n = 1; no nonzero-base hypothesis is imposed. The difference Ω − ω is natural subtraction, equal to the ordinary difference because ω ≤ Ω. Other OEIS assertions are outside the claim.

Verified locator

• URL: https://oeis.org/A001222
• A001222 %N (verbatim): Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).
• A001222 %O (verbatim): 1,3
• A001222 %C (verbatim): Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - Werner Schulte, Feb 10 2018
• A046660 URL: https://oeis.org/A046660
• A046660 %N (verbatim): Number of prime factors of n counted with multiplicity minus number of distinct prime factors. Omega(n) - omega(n).
• A001221 URL: https://oeis.org/A001221
• A001221 %N (verbatim): Number of distinct primes dividing n (also called omega(n)).
• Dressler and van de Lune, Proc. AMS 41 (1973), 403–406, doi 10.1090/S0002-9939-1973-0340191-8; proves only the case x = 1 − y.

Readings of 2026-09-16: the OEIS entry still marks the line Conjecture and cites only the x = 1 − y case as proved; OpenAlex (3 unrelated hits), Math.SE (0) and formal-conjectures (0) show no proof of the general case; arXiv was not searched (no API response). Historical openness beyond these surfaces is ASSUMED-UNVERIFIED.