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bibkey: laboselemer2013a046528 authors: Labos Elemer; Jaroslav Krizek year: 2013 title: “OEIS A046528, Numbers that are a product of distinct Mersenne primes” doi: null url: https://oeis.org/A046528 claim: “A046528 %N: Numbers that are a product of distinct Mersenne primes (elements of A000668). A046528 %C: n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit. A046528 %C: 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” strata_touched:

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

OEIS A046528

The NAME of A046528 states:

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

The known COMMENT 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 of May 4, 2013 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 AUTHOR line is:

Labos Elemer

The AUTHOR line carries no date. The metadata year 2013 is the date of Krizek’s conjecture, while Labos Elemer remains first as the sequence author.

Verified locator

  • URL: https://oeis.org/A046528
  • NAME (verbatim): Numbers that are a product of distinct Mersenne primes (elements of A000668).
  • Known COMMENT (verbatim): n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.
  • Krizek COMMENT (verbatim): 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 known classification of the integers with sigma equal to a power of two is an attributed prerequisite. The OEIS %D and %H lines point to R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Dekker, 1989; Jeffrey Shallit, Problem 1319, Mathematics Magazine 63 (1990), 129; and C. D. H. Cooper, Problem E 2493, American Mathematical Monthly 81 (1974), 902, with W. J. Dodge’s solution in volume 82 (1975).

Only Krizek’s rational-power equivalence is resolved here. It is stated in the integer-power form sigma(n)^b = tau(n)^a, equivalent for these positive integer bases to sigma(n) = tau(n)^(a/b).