Problems and Solutions

Callan, David; Lemma, Mulatu; Lim, Joung-kuen; Lee, Ho-joo; Hemlow, Lance E.; Marquez, Juan-Bosco Romero; Holshouser, Arthur L.; Kinyon, Michael; Kashihara, Kenichiro; Martin, Paul; Trench, William F.; Andreoli, Michael H.; Spence, Lawrence; Eynden, Charles Vanden; Sumner, John S.; Dove, Kevin L.; Graham‐Eagle, James; Callender, Kennard

doi:10.2307/2687105

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Superharmonic Numbers1

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2009

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“…The proof by Callan [2] (see also Pomerance [9]) that the only harmonic numbers with two distinct prime factors are the even perfect numbers does not carry through to superharmonic numbers although it would seem to be true.…”

Section: Superharmonic Numbersmentioning

confidence: 99%

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Superharmonic numbers

Cohen

2009

Math. Comp.

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Abstract. Let τ (n) denote the number of positive divisors of a natural number n > 1 and let σ(n) denote their sum. Then n is superharmonic if σ(n) | n k τ (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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Section: Superharmonic Numbersmentioning

confidence: 99%

“…In fact, it it is enough to show that (2) #{n ≤ x : n ∈ P 2 , q = P + (n) > y, q n, P + (q + 1) > z} = o(x),…”

Section: Theoremmentioning

confidence: 99%