Superharmonic numbers

Let \tau (n) denote the number of positive divisors of a natural number n>1 and let \sigma (n) denote their sum. Then n is \emph {superharmonic} if \sigma (n)\mid n^k\tau (n) for some positive integer k. We deduce numerous properties of superharmonic numbers and show in particular that the set of...

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Bibliographic Details
Published in:Mathematics of computation 2009-01, Vol.78 (265), p.421-429
Main Author: Cohen, Graeme L.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Harmonic mean
Index numbers
Integers
Mathematical integrals
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Natural numbers
Numbers
Numerical analysis
Numerical analysis. Scientific computation
Prime numbers
Quotients
Sciences and techniques of general use
Superharmonics
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Summary:Let \tau (n) denote the number of positive divisors of a natural number n>1 and let \sigma (n) denote their sum. Then n is \emph {superharmonic} if \sigma (n)\mid n^k\tau (n) for some positive integer k. We deduce numerous properties of superharmonic numbers and show in particular that the set of all superharmonic numbers is the first nontrivial example that has been given of an infinite set that contains all perfect numbers but for which it is difficult to determine whether there is an odd member.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-08-02147-9