A quick glance at the list of the first $ k $ -multiperfect numbers for small $ k $ makes me think that all $ k $ -multiperfect numbers are multiples of $ k $ , which is a generalization of the famous "no odd perfect number" conjecture. Has this conjecture appeared so far ? If yes, what are the results towards it so far ?

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    $\begingroup$ Wikipedia says it's an open question whether $k$-perfect numbers are divisible by $k!$. $\endgroup$ – Qiaochu Yuan Dec 3 '18 at 23:29
  • $\begingroup$ @QiaochuYuan: By chance, do you happen to know of any reference (in the literature) to the conjecture that $k$-perfect numbers must be divisible by $k!$? I checked the Wikipedia page and there is none there. $\endgroup$ – Jose Arnaldo Bebita-Dris Dec 19 '18 at 5:35

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