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I am looking for pairs of positive integers $(m,n)$ such that $ \sigma(n!-1) =m^2$, where $\sigma$ is the sum of divisors function. Examples occur with $(m,n)=(12,5),(1,2)$.

Question: Are there others?

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    $\begingroup$ I can't help my curiosity. Could you say more about your problem? $\endgroup$
    – Włodzimierz Holsztyński
    Commented Nov 24, 2016 at 19:57
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    $\begingroup$ No more solutions with $n<60$. $\endgroup$
    – Wojowu
    Commented Nov 24, 2016 at 20:46
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    $\begingroup$ Sure. It's not hard to come up with open problems in number theory. Some probabilistic argument will say something, and then we get stuck. $\endgroup$
    – znt
    Commented Nov 24, 2016 at 22:38
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    $\begingroup$ Numbers $k$ such that $\sigma(k)$ is a square are tabulated at oeis.org/A006532 – it is not even known that there are infinitely many of them (although this would follow from standard conjectures). $\endgroup$
    – Gerry Myerson
    Commented Nov 25, 2016 at 4:31
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    $\begingroup$ @Gerry, the OEIS article of Buekers et al claim there are infinitely many n whose sigma value is a square. Gerhard "From Looking At First Page" Paseman, 2016.11.25. $\endgroup$
    – Gerhard Paseman
    Commented Nov 25, 2016 at 20:47

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