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Number of distinct near-squares primes dividing an odd perfect number

In general, very few prime factors in an odd perfect number can be of the form $n^2+1$. In particular, if N is an odd perfect number then $\frac{\sigma(N)}{N}=2$, and for any $m$ (perfect or not), $\...
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If $p^k m^2$ is an odd perfect number with special prime $p$, then under what other conditions on $\sigma(p^k)/2$ does $k=1$ follow?

This is a partial answer, in line with my first inquiry. It turns out that the converse $$k = 1 \implies \sigma(p^k)/2 \text{ is squarefree }$$ is true. The proof that follows is lifted from this ...
1 vote
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Divisibility relation with a specific sum of divisors

From the paper of Touchard that is linked in the question on we get the relation $$3nS_0(n)-\frac{n(n-1)\sigma(n)}{6}=\frac{10}{n}S_2(n) ....(1)$$ here $S_i(n)=\sum_{k=1}^{n-1}k^i\sigma(k)\sigma(n-k)$....
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