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2 results

  • ON ODD PERFECT NUMBERS

  • Part of
  • FENG-JUAN CHEN, YONG-GAO CHEN
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 86 / Issue 3 / December 2012
    Published online by Cambridge University Press:
    16 February 2012, pp. 510-514
    Print publication:
    December 2012
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  • Let q be an odd prime. In this paper, we prove that if N is an odd perfect number with qα∥N then σ(N/qα)/qα≠p,p2,p3,p4,p1p2,p21p2, where p,p1, p2 are primes and p1≠p2. This improves a result of Dris and Luca [‘A note on odd perfect numbers’, arXiv:1103.1437v3 [math.NT]]: σ(N/qα)/qα≠1,2,3,4,5. Furthermore, we prove that for K≥1 , if N is an odd perfect number with qα ∥N and σ(N/qα)/qα ≤K, then N≤4K8.

  • BOUNDS FOR ODD k-PERFECT NUMBERS

  • Part of
  • SHI-CHAO CHEN, HAO LUO
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 84 / Issue 3 / December 2011
    Published online by Cambridge University Press:
    21 July 2011, pp. 475-480
    Print publication:
    December 2011
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  • Let k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is bounded by (k−1)4r3.