Euler's factor and the order of odd perfect numbers (Q2739213)

A theorem of Euler states that an odd perfect number must have the form \(p^\alpha p_1^{2\alpha_1}\cdots p_s^{2\alpha_s}\), where \(p\) and \(p_i\) are distinct odd primes and \(p\equiv \alpha\equiv 1\pmod 4\). \textit{P. Starni} [J. Number Theory 37, 366-369 (1991; Zbl 0721.11001)] showed that if \(p_i\equiv 1\pmod 4\) for all \(1\leq i\leq s\), then \(p\equiv \alpha\pmod 8\). The author further shows that if \(k\) denotes the number of \(p_i\) satisfying \(p_i\equiv 1\pmod 4\) and \(2\nmid \alpha_i\), then \(p-\alpha\equiv 0\) or \(4\pmod 8\), depending on whether \(2\mid k\) or not.