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?

  • 6
    $\begingroup$ I can't help my curiosity. Could you say more about your problem? $\endgroup$ – Włodzimierz Holsztyński Nov 24 '16 at 19:57
  • 1
    $\begingroup$ No more solutions with $n<60$. $\endgroup$ – Wojowu Nov 24 '16 at 20:46
  • 8
    $\begingroup$ 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 Nov 24 '16 at 22:38
  • 4
    $\begingroup$ 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 Nov 25 '16 at 4:31
  • 2
    $\begingroup$ @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 Nov 25 '16 at 20:47

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.