On necessary conditions for the existence of odd perfect numbers (Q1293287)

It is an open question, whether there exists an odd perfect number. L. Euler has shown, that such numbers must be of the form \(4m+1\). The author proves, that \[ (4m+1)^2+ \sum_{k=1}^{2m} \sigma(8m+3-2k) \sigma(2k-1) \] must be even and that \[ \#\{k\in \mathbb N: k\leq 2n,\;8m+3-2k \text{ is a square and \(2k-1\) is square}\} \] must be odd. His method uses generalizations of Ramanujan identities.