On some properties of the Euler's factor of certain odd perfect numbers

Abstract

Let n = 3 2 Q 2 be an odd positive integer, with prime, ≡ ≡ 1 (mod 4), Q squarefree, (Q, ) = (Q, 3) = 1. It is shown that: if n is perfect, then ( ) ≡ 0 (mod 3 2 ). Some corollaries concerning the Euler's factor of odd perfect numbers of the above mentioned form, if any, are deduced.

Odd Multiperfect Numbers

“…Based on a result of McDaniel [7], Starni [10] proved that if an odd 2-perfect number n has the form n = π α 3 2β Q 2β with (3, Q) = 1, then 3 2β |σ(π α ). This result was generalized to odd 2 k -perfect numbers by Broughan and Zhou [2, Theorem 2.6].…”

“…It turns out that the properties of the Euler part of k-perfect numbers can be used to prove the nonexistence of odd k-perfect numbers of certain forms. For instance, Starni [11] recently proved that if an odd perfect number n has the form n = π α 3 2β Q 2β with (3, Q) = 1, then 3 2β | σ(π α ). As a corollary, he showed that if π ≡ 1 (mod 12) and α ≡ 1 or 9 (mod 12), then there do not exist odd perfect numbers n = π α 3 2β M 2 .…”

Odd multiperfect numbers of abundancy 4

“…The following result is based on the technique of Starni [6] whose theorem, for 2-perfect numbers, had uniform powers for the p i . This, in turn depended on a result of McDaniel [3] (incorrectly cited as [4]), where the powers are not uniform.…”

New Congruences for Odd Perfect Numbers

“…[18] If n = p α 3 2β d 2β with square-free d and with gcd(3, d) = 1 = gcd(p, d) is an odd perfect number then σ(p α ) ≡ 0 (mod 3 2β ).Lemma 2.5. [20, footnote on page 590]There is no odd perfect number n divisible by 105 = 2 • 5 • 7.…”