TY  - JOUR
AB  - We say n ? ? is perfect if ? (n) = 2n, where ?(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? ?kj=1 q2?jj , where p, q1, ?, qk, are distinct primes and p ? ? ? 1 (mod 4). We prove that if ?j ? 1 (mod 3) or ?j ? 2 (mod 5) for all j, 1 ? j ? k, then 3 ? n. We also prove as our main result that ?(n) ? 37, where ?(n) = ? + 2?kj=1 ?j. This improves a result of Sayers (?(n) ? 29) given in 1986.
AU  - Iannucci, DE
AU  - Sorli, RM
DA  - 2003/01/01
DO  - 10.1090/S0025-5718-03-01522-9
EP  - 2084
JO  - Mathematics of Computation
PY  - 2003/01/01
SP  - 2077
TI  - On the total number of prime factors of an odd perfect number
VL  - 72
Y1  - 2003/01/01
Y2  - 2026/05/01
ER  -