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slug: oeis-a046528-sigma-tau-rational-power-mersenne-characterization bibkey: laboselemer2013a046528 doi: null url: https://oeis.org/A046528 triage: theorem motivation_gids:

  • D5/S3/Arith/Mersenne/KrizekSigmaTauRationalPowerMersenne

Products of distinct Mersenne primes satisfy exactly the sigma-tau rational-power relation

Problem

OEIS A046528, NAME (%N, verbatim):

Numbers that are a product of distinct Mersenne primes (elements of A000668).

The known COMMENT (%C, verbatim) states:

n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.

Jaroslav Krizek’s COMMENT (%C, verbatim) states:

Supersequence of A051281 (numbers n such that sigma(n) is a power of tau(n)). Conjecture: numbers n such that sigma(n) = tau(n)^(a/b), where a, b are integers >= 1. Example: sigma(93) = 128 = tau(93)^(7/2) = 4^(7/2). - Jaroslav Krizek, May 04 2013

The literal proved statement is ∀ n : ℕ, 0 < n → (isMersenneProduct n ↔ ratPow n), where isMersenneProduct n means ∃ S : Finset ℕ, n = ∏ p ∈ S, p ∧ ∀ p ∈ S, p.Prime ∧ ∃ k : ℕ, 0 < k ∧ p + 1 = 2 ^ k, and ratPow n means ∃ a b : ℕ, 0 < a ∧ 0 < b ∧ (σ 1 n) ^ b = (σ 0 n) ^ a. Here σ 1 is the divisor-sum function and σ 0 is the divisor-count function tau. Since both bases are positive integers for positive n, this integer-power equality is equivalent to Krizek’s rational-exponent form.

No statement about A051281 beyond the displayed equivalence is claimed. The infinitude or distribution of Mersenne primes and novelty of the known sigma-equals-a-power-of-two classification are also NOT claimed.

Motivation

The conjecture gives a structural characterization of every positive integer whose divisor sum and divisor count are related by positive rational powers. It strengthens the practical description of A046528 from a factorization condition to an equivalent relation between two standard arithmetic functions.

Gap

The OEIS %C line still labels Krizek’s assertion as a conjecture dating from 2013. The checked A046528 history through revision 85 contains no settlement, and the A051281 discussion includes Marcus’s 2020 remark without a proof of this equivalence. Exact searches performed on 2026-09-14 returned zero arXiv matches, zero OpenAlex matches, and zero GitHub exact-phrase matches. MathOverflow question 62721 concerns only the known equation sigma(x) = 2^n, not Krizek’s sigma-tau equivalence. These checked surfaces do not support a publication-priority claim.

Route

  1. Equality of positive powers gives equality of the prime supports of sigma(n) and tau(n).
  2. For the largest odd prime q dividing tau(n), choose a prime-power factor p^e of n with q ∣ e+1; then the geometric sum G = ∑ i < q, p^i divides sigma(n).
  3. Every prime factor r of G equals q: support equality gives r ≤ q, while the multiplicative order of p modulo r gives either r=q or the contradiction q < r.
  4. The theorem emultiplicity_geom_sum₂_eq_one gives q^2 ∤ G, whereas the preceding unique-prime-factor classification forces the contrary. Thus tau(n) is a power of two. Steps 1 through 4 are the new argument.
  5. The known Sivaramakrishnan-Shallit prerequisite, proved inline and attributed, shows that simultaneous power-of-two values of tau(n) and sigma(n) force n to be a product of distinct Mersenne primes.
  6. Conversely, multiplicativity gives sigma(n)=2^(sum k_p) and tau(n)=2^|S|; the empty product n=1 uses exponents a=b=1.

Falsifier

A positive natural number n for which exactly one of isMersenneProduct n and ratPow n holds would contradict the theorem. The kernel-checked result quantifies over every positive natural number.

Evidence

  • Lean module: D5/S3/Arith/Mersenne/KrizekSigmaTauRationalPowerMersenne.lean.
  • Main theorem: result, with std3 axiom closure [propext, Classical.choice, Quot.sound].
  • Kernel profile on this worktree: wall time 6.04 seconds, type checking 160 milliseconds, and maximum resident set size 2,208,792,576 bytes.
  • The orchestrator independently found zero counterexamples to the equivalence for n ≤ 10^5, zero counterexamples to the key odd-prime-support lemma for n ≤ 2 * 10^5, and zero counterexamples to the known step 5 over the same latter range.
  • The probe found 19 ratPow hits for n ≤ 10^5; every hit was a product of distinct Mersenne primes.
  • The bounded checks support fault detection only; the Lean proof carries the universal result.

Triage

theorem. Krizek’s rational-power characterization is proved for every positive natural number, so the resolution is proved.

ASSUMED-UNVERIFIED

The bounded scans do not establish the universal statement. The source texts of Sivaramakrishnan and Shallit were not opened; their attribution is taken from the OEIS %C line. Historical openness outside the checked OEIS history, arXiv, OpenAlex, MathOverflow, GitHub, and cited-source surfaces is unverified; no exhaustive literature or priority claim is made.