On some properties of the Euler's factor of certain odd perfect numbers (Q818072)

Let \(\sigma(n)\) be the sum of the positive divisors of the natural number \(n\); \(n\) is said to be perfect if \(\sigma(n)=2n\). It is shown that if an odd perfect integer \(n\) is of the form \(p^a(3Q)^{2b}\) with \(p\), \(a\), \(Q\) and \(b\) satisfying certain restrictions, then \(\sigma(p^a)\equiv 0 \pmod {3^{2b}}\). The proof is elementary.