bibkey: schulte2018a322327 authors: Werner Schulte year: 2018 title: “OEIS A322327, A005361(n)·A034444(n), with the conjecture that A005361(n)·k^ω(n) is multiplicative with prime-power values k·e” doi: null url: https://oeis.org/A322327 claim: “%N a(n) = A005361(n) * A034444(n). %C Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.” strata_touched:

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

OEIS A322327

Schulte’s conjecture concerns the function obtained by multiplying the prime-factorization exponents and then multiplying by an integer parameter raised to the number of distinct prime divisors. The formal scope is the two assertions in the quoted conjecture sentence for every integer k: multiplicativity on coprime positive arguments and the prime-power value k*e. The trailing sequence correspondences are outside the claim. Lean’s natural-power convention supplies 0^0 = 1.

Verified locator

• URL: https://oeis.org/A322327
• A322327 %N (verbatim): a(n) = A005361(n) * A034444(n).
• A322327 %O (verbatim): 1,2
• A322327 %A (verbatim): Werner Schulte, Dec 03 2018
• A322327 %C (verbatim): Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.
• A005361 %N (verbatim): Product of exponents of prime factorization of n.
• A001221 %N (verbatim): Number of distinct primes dividing n (also called omega(n)).

Readings of 2026-09-16: the OEIS entry still marks the line Conjecture (last edited 2026-04-09) with no proof line; OpenAlex "A322327" 0 hits; Math.SE API 0 hits; arXiv was not searched (no API response). Historical openness beyond these surfaces is ASSUMED-UNVERIFIED.