More on the total number of prime factors of an odd perfect number

Abstract

Let σ ( n ) \sigma (n) denote the sum of the positive divisors of n n . We say that n n is perfect if σ ( n ) = 2 n \sigma (n) = 2 n . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p α ∏ j = 1 k q j 2 β j N = p^\alpha \prod _{j=1}^k q_j^{2 \beta _j} , where p , q 1 , … , q k p, q_1, \ldots , q_k are distinct primes and p ≡ α ≡ 1 ( mod 4 ) p \equiv \alpha \equiv 1 \pmod {4} . Define the total number of prime factors of N N as Ω ( N ) := α + 2 ∑ j = 1 k β j \Omega (N) := \alpha + 2 \sum _{j=1}^k \beta _j . Sayers showed that Ω ( N ) ≥ 29 \Omega (N) \geq 29 . This was later extended by Iannucci and Sorli to show that Ω ( N ) ≥ 37 \Omega (N) \geq 37 . This paper extends these results to show that Ω ( N ) ≥ 47 \Omega (N) \geq 47 .