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(I have asked a similar question in MSE around a week ago, but did not receive any responses. I have therefore cross-posted it to this site, hoping to get some answers.)

An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime, $q \equiv k \equiv 1 \pmod 4$, and $\gcd(q, n) = 1$.

Here is my question:

If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ squarefree?

I would certainly appreciate it if someone could point me to existing papers in the literature where this particular question is addressed.

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No, if an odd perfect number exists, then $n$ must contain a square factor. This is a 1937 result of Steuerwald:

R. Steuerwald, "Verschärfung einer notwendigen Bedingung für die Existenz einer ungeraden vollkommenen Zahl," S.-B. Math.-Nat. Abt. Bayer. Akad. Wiss., 1937, pp. 68-73.

A very nice recent paper on this theme is by Fletcher, Nielsen, and Ochem (Math. Comp.); the preprint version is freely available on Nielsen's website: https://math.byu.edu/~pace/OPNSieves_web.pdf

[Edited as per Lucia's suggestion.]

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  • $\begingroup$ It is absurd that someone chose to vote down this answer! (It took me a little effort to parse your answer -- maybe it would be clearer if the first line said something like: ``No: if an odd perfect number exists, then $n$ must necessarily contain a square factor.") $\endgroup$
    – Lucia
    Commented Dec 18, 2015 at 22:33
  • $\begingroup$ Yes, that's clearer. I've made the edit.Thanks! $\endgroup$
    – so-called friend Don
    Commented Dec 19, 2015 at 4:32
  • $\begingroup$ @so-calledfriendDon, mind if you check out this related MSE question, where I ask if $n$ is a square? Let me know if you need me to ask a separate MO question. $\endgroup$
    – Jose Arnaldo Bebita
    Commented Dec 24, 2015 at 14:10
  • $\begingroup$ @so-calledfriendDon, here is the new MO question. $\endgroup$
    – Jose Arnaldo Bebita
    Commented Dec 27, 2015 at 13:08

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