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Looking for an appropriate reference(s) for two conjectures on odd perfect numbers
(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)
Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For ...
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Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture
It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
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If $N = q^k n^2$ is an odd perfect number, and $n < q^{k+1}$, does it follow that $k > 1$?
Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. According to Dickson (as pointed out recently by Beasley), Descartes conjectured $k=1$ in a letter to Mersenne in 1638, with Frenicle'...