[seqfan] A280098 (A018253)

Don Reble djr at nk.ca
Wed Mar 25 08:42:35 CET 2020


> %C A280098 Conjecture: only the integers k in {1, 3, 4, 6, 8, 12, 24}
>    have the property that [the sum of the divisors of (k*n-1)]/k is
>    always an integer. - Robert G. Wilson v, Dec 25 2016

    He's right.
    My proof relates to self-inverse multiplicative groups;
    those moduli are {2,3,4,6,8,12,24} (almost Wilson's set).

    A018253 refers to many proofs, that those are the self-inverse groups.
    One can easily prove that, using the fact that the map "x -> x mod m"
    from any modulo-km group to the modulo-m group is onto.
    But then I struggled to prove that fact. I finally got it:
    enumerate the kernel of the mapping, compute the number of cosets.
    But it seems like swatting a mosquito with an anvil.
    Did I overlook something simple?

Don Reble  djr at nk.ca

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