Let q be an odd prime. In this paper, we prove that if N is an odd perfect number with qα∥N then σ(N/qα)/qα≠p,p2,p3,p4,p1p2,p21p2, where p,p1, p2 are primes and p1≠p2. This improves a result of Dris and Luca [‘A note on odd perfect numbers’, arXiv:1103.1437v3 [math.NT]]: σ(N/qα)/qα≠1,2,3,4,5. Furthermore, we prove that for K≥1 , if N is an odd perfect number with qα ∥N and σ(N/qα)/qα ≤K, then N≤4K8.