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IS2003

D. E. Iannucci and R. M. Sorli, "On the total number of prime factors of an odd perfect number," Math. Comp., 72:244 http://www.ams.org/jourcgi/jour-getitem?pii=S0025-5718-03-01522-9, (2003) 2077--2084.  MR 1986824

Abstract: We say n∈ N is perfect if σ(n)=2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n=p^α∏_j=1^kq_j^2β_j, where p, q₁, ..., q_k are distinct primes and p≡α≡1 (mod 4) . We prove that if β_j≡1 (mod 3) or β_j≡2 (mod 5) for all j, 1≤ j≤ k, then 3 n. We also prove as our main result that Ω(n)≥37, where Ω(n)=α+2∑_j=1^kβ_j. This improves a result of Sayers (Ω(n)≥ 29) given in 1986.