On odd perfect, quasiperfect, and odd almost perfect numbers

Abstract

We establish upper bounds for the six smallest prime factors of odd perfect, quasiperfect, and odd almost perfect numbers.

On the distribution of some integers related to perfect and amicable numbers

“…Moreover, 2 1 n 1 by condition (iv), so that 1 + 1 = σ( 1 ) | σ(n 1 ). Set r := P ( 1 + 1); (15) recalling condition (vi), we find that…”

“…Here r, 1 , 2 , and P 1 are understood to be prime; the congruence conditions on 1 , 2 follow from (15) and (16), the congruence m 1 ≡ 0 (mod 1 ) comes from (14), and the inequality P 1 ≥ 2 follows from our initial assumption that P (n 1 ) ≥ P (n 2 ). Making crude upper estimates at each step, we find that our quintuple sum is…”

“…Any such example must be an odd square [4], must possess at least seven distinct prime factors, and must have more than than 35 decimal digits (for these last two results, see [9]). For other theoretical work on quasiperfect numbers, see [1,22,5,15,6,14]. About quasiamicable pairs, almost all of what we know has been gleaned from computer searches [16,10,3].…”

On the small prime factors of a non-deficient number

“…We prove similar bounds with p 4 when n is odd and perfect, and a slightly weaker result when n is a primitive non-deficient number. Using the same techniques we will also produce estimates for σ(m) when m is near but not equal to n!.Kishore [5] proved that if n is an odd perfect number with k distinct prime factors, and 2 ≤ i ≤ 6, then one must haveNote that these bounds are all linear in p i and k. In this paper, for i in the range 2 ≤ i ≤ 4, we prove improve on Kishore's results, giving better than linear bounds. Moreover, for i = 2 and i = 3, our results apply not just to odd perfect numbers but also to odd abundant numbers.For p 1 , better than linear bounds are known, due to [9] and the author [13].…”

“…Kishore [5] proved that if n is an odd perfect number with k distinct prime factors, and 2 ≤ i ≤ 6, then one must have…”