On the number of distinct prime factors of an odd perfect number

Sieve methods for odd perfect numbers

“…For 2 i 6, Kishore [20] showed that p i < 2 and this has been slightly improved by Cohen and Sorli [3].…”

Odd perfect numbers have at least nine distinct prime factors

“…We will use a computer run algorithm to prove our results. The main idea of the algorithm is to use a factor chain argument, as given in both [5] and [25]. For the benefit of the reader, we will describe the algorithm here.…”

“…(This corresponds to the case when ∆ 0 > 2 in Lemma 20.) If we make B large enough, the intervals will always have upper bounds (for a proof see [5]), and the algorithm will only have to consider a finite number of cases.…”

“…There are three major differences between our implementation of this algorithm, and the implementations in [5] and [25]:…”

On the total number of prime factors of an odd perfect number

“…A related problem is that of finding a lower bound on Ω(n) for an odd perfect number n. The first significant result of this type was obtained by Cohen [3] in 1982 when he proved that Ω(n) ≥ 23. In 1986, Sayers [12] improved this result to obtain Ω(n) ≥ 29.…”