Superharmonic Numbers

Amer Mathematical Soc
Publication Type:
Journal Article
Mathematics Of Computation, 2008, 78 (265), pp. 421 - 429
Issue Date:
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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 superharmonic if sigma(n) vertical bar 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.
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