# When is $\sigma(n!-1)$ a perfect square?

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?

• I can't help my curiosity. Could you say more about your problem? – Włodzimierz Holsztyński Nov 24 '16 at 19:57
• No more solutions with $n<60$. – Wojowu Nov 24 '16 at 20:46
• 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. – znt Nov 24 '16 at 22:38
• 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). – Gerry Myerson Nov 25 '16 at 4:31
• @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. – Gerhard Paseman Nov 25 '16 at 20:47