Summary: Let \(N\) be an odd perfect number and let \(a\) be its third largest prime divisor, \(b\) be the second largest prime divisor, and \(c\) be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on \(a\), as well as the closely related problem of improving bounds for \(bc\) and \(abc\). In particular, we prove two results. First, we prove a new general bound on any prime divisor of an odd perfect number 1 and obtain as a corollary of that bound that \(a<2N^{\frac{1}{6}}\). Second, we show that \(abc <(2N)^{\frac{3}{5}}\). We also show how in certain circumstances these bounds and related inequalities can be tightened. Define a \(\sigma_{m,n}\) pair to be a pair of primes \(p\) and \(q\) where \(q| \sigma (p^m)\) and \(p| \sigma (q^n)\). Many of our results revolve around understanding \(\sigma_{2,2}\) pairs. We also prove results concerning \(\sigma_{m,n}\) pairs for other values of \(m\) and \(n\).