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Prime-Perfect Numbers

Paul Pollack / Carl Pomerance
Published Online: 2012-11-30 | DOI: https://doi.org/10.1515/integers-2012-0044


We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count of prime-perfect numbers in satisfies estimates of the form

as . We also discuss the analogous problem for the Euler function. Letting denote the number of for which n and share the same set of prime factors, we show that as ,

We conclude by discussing some related problems posed by Harborth and Cohen.

Keywords: Sum-of-Divisors Function; Perfect Number; Harmonic Number; Superharmonic Number

About the article

Received: 2011-01-11

Accepted: 2011-06-13

Published Online: 2012-11-30

Published in Print: 2012-12-01

Citation Information: Integers, Volume 12, Issue 6, Pages 1417–1437, ISSN (Online) 1867-0652, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integers-2012-0044.

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© 2012 by Walter de Gruyter Berlin Boston.Get Permission

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