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On odd perfect numbers (II), multiperfect numbers and quasiperfect numbers

  • Graeme L. Cohen (a1)
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Let N be a positive integer. This paper is concerned with obtaining bounds for (p prime), when N is an odd perfect number, a multiperfect number, or a quasiperfect number, under assumptions on the existence of such numbers (where none is known) and whether 3 and 5 are divisors. We argue that our new lower bounds in the case of odd perfect numbers are not likely to be significantly improved further. Triperfect numbers are investigated in some detail, and it is shown that an odd triperfect number must have at least nine distinct prime factors.

1980 Mathematics subject classification (Amer. Math. Soc.): 10 A 20.

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References
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Abbott, H. L.Aull, Ezra Brown and D. Suryanarayana (1973), 'Quasiperfect numbers', Ada Arith. 22, 439447; (1976), 'Corrections to the paper “Quasiperfect numbers”', Acta Arith. 29, 427–428.
Cohen, G. L. (1978), 'On odd perfect numbers', Fibonacci Quart 16, 523527.
Kishore, Masao (1978), 'Odd integers N with five distinct prime factors for which 2–10–12Math. Comp. 32, 303309.
Krawczyk, Andrzej (1972), 'Uogo1nienie problemu Perisastriego', Prace Nauk. Inst. Mat. Fiz. Teoret. Politechn. Wroclaw. Ser. Studia i Materialy No. 6 Zagadnienia kombinatoryczne, 6770 (English summary).
Suryanarayana, D. (1963), 'On odd perfect numbers II', Proc. Amer. Math. Soc. 14, 896904.
Suryanarayana, D. and Peter Hagis, Jr. (1970), 'A theorem concerning odd perfect numbers', Fibonacci Quart. 8, 337346, 374.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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