Some results concerning quasiperfect numbers

Abstract

New methods are introduced here to show that if n is a quasiperfect number and u(n) the number of its distinct prime factors, then u(n) > 7 and n > 10 35 , and if further 3}« then u(n) > 9 and n > 1O 40 .

Untitled

“…Although this theorem was new to us, we have learned that this is not the first time it has appeared in the literature. The referee pointed out that this theorem appears in more general form in [Pomerance 1975], from which we learned that the first appearance of Theorem 2 was in a note by Mąkowski [1960].…”

“…Keywords: sigma function, sum of divisors, excedents, computational mathematics. (1) n D k 2 , where k is odd [Cattaneo 1951]; (2) if m is a proper divisor of n, then .m/ < 2n [Cattaneo 1951]; (3) if r j .n/ then r Á 1 or 3 .mod 8/ [Cattaneo 1951]; (4) n has at least seven prime factors [Hagis and Cohen 1982];…”

On the problem σod(n) = σod(n+1)

“…Therefore, t must be odd. If p ≡ −1 (mod 4), then p t + 1 ≡ 0 (mod 4) as t is odd, which contradicts to (12). So p ≡ 1 (mod 4).…”

“…But still none of them is found. Hagis and Cohen [12] proved ω(n) ≥ 7 for n being a quasi-perfect number.…”

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

“…No quasiperfect numbers are known. 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].…”