On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime

Abstract

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{{\alpha }_1}\cdots p_s^{{\alpha }_s}{q}_1^{{\beta }_1}\cdots q_t^{{\beta }_t}$ with ${\beta }_1,\dots ,{\beta }_t$ bounded by a given constant. We shall also give similar results for quasiperfect numbers and relatively prime amicable pairs of opposite parity.

original: arXiv:math/0511410; 11 pages, retitled and included some new results; [v2] 16 pages, rearraged the argument and changed some constants; [v3] 17 pages, some revisions, [v6] 17 pages, the author's final version, address changed due to the location change of the Minoh Campus of Osaka University

References in corpus (2)

• An explicit zero-free region for the Dirichlet L-functions
• Explicit formulae for primes in arithmetic progressions, I