About A066289

[Don Reble]

Seqfans, Thomas Baruchel, Rich Schroeppel:


A month ago, Thomas Baruchel wrote:
> Hi, I found more terms for
>
> %I A066289
> %S A066289 1,6,120,672,30240,32760,31998395520
> %N A066289 Numbers n such that Mod[DivisorSigma[2k-1,n],n]=0
>             holds presumably for all k;
> i.e. all odd-power-sum of divisors of n are divisible by n.
> %F A066289 DivisorSigma[2k-1,n]/n] is integer for all k=1,2,3,..,200,..
> %e A066289 Tested for each n and k<200.
> Otherwise the proof for all k seems laborious, ...
>
> these terms are:
>   * 796928461056000
>   * 212517062615531520
>   * 680489641226538823680000
>   * 13297004660164711617331200000
>
> But I don't know if they come exactly after the last current term,
> and i don't know either if no term is missing between them.
> Is there something (greatness ???) that would make you think they
> come exactly after the last current term with no missing term
> between them ?


Luckily, the proofs aren't very laborious. Of course, testing up to
k=200 provides a quick refutation for most numbers. For those numbers N
that pass, one must check, for each prime-power PP that divides N,
whether PP also divides the odd-power-sum (OPS) of divisors.

One can exploit the multiplicativity of that OPS function, and the fact
that modulo PP, it's periodic. For all the numbers I tested, either
    - PP is small, and one can test whether PP divides OPS throught the
      whole period, or
    - PP is large, but for each other prime-power-factor PP2 of N,
      one can find a small factor SF of PP, such that the combined
      contributions of each PP2 (modulo its SF) are enough to show that
      PP divides OPS.

Of course, I tested the multiperfect numbers at
    http://wwwhomes.uni-bielefeld.de/achim/mpn.html
except for the last two megadigit numbers, 2^6972592.M6972593 and
2^13466916.M13466917. (Insufficient memory)-:

Of those numbers, I find that only these are in A066289:

1 6 120 672 30240 32760 31998395520 796928461056000
212517062615531520 680489641226538823680000
13297004660164711617331200000 1534736870451951230417633280000
6070066569710805693016339910206758877366156437562171488352958895095808000000000

(Hey, Thomas, we agree!)

But I wonder, does that list have all multiperfect numbers up to
6070066...? Rich, do you know?

--
Don Reble       djr at nk.ca