[seqfan] Re: New "metaperfect" number for A068978

[Allan Wechsler]

In a situation like this, where we are losing confidence in how exhaustive
the list is, should we retain the index-number column? If we keep index
numbers, all the entries downstream from the new ones will change indices.

On Thu, Mar 28, 2024 at 10:24 PM Neil Sloane <njasloane at gmail.com> wrote:

> Allan,
> >  Perhaps we should change the
> text of the entry so that this file is labeled "Other examples, not
> necessarily consecutive", and add my new discovery to it?
>
>
> Sounds good!  You can download the list from the entry, then add your
> value(s), and resubmit it.
>
> Best regards
> Neil
>
> Neil J. A. Sloane, Chairman, OEIS Foundation.
> Also Visiting Scientist, Math. Dept., Rutgers University,
> Email: njasloane at gmail.com
>
>
>
> On Thu, Mar 28, 2024 at 8:34 PM Allan Wechsler <acwacw at gmail.com> wrote:
>
> > The sequence oeis.org/A007429 records the sum, over all divisors of n,
> of
> > sigma_1(n).
> >
> > Sigma_1 itself (oeis.org/A000203) records the sum of the divisors
> > themselves.
> >
> > This "nested sigma" calculation causes me to think of A007429 as the
> > "metasigma" function. Like sigma_1, it is multiplicative. The basis can
> > easily be seen to be the following:
> >
> > A007429(p^k) = p^k + 2 p^(k-1) + 3 p^(k-2) + ... + (k+1)
> >
> > where the coefficient and the exponent always add to k+1.
> >
> > This "sigma-like" function gives rise to an analog of the multiperfect
> > numbers, which I think of as "metaperfect". A number N is metaperfect if
> N
> > divides A007429(N). These numbers are recorded in oeis.org/A068978. The
> > entry gives the first 28 examples in the data, found by Benoit Cloitre,
> > Rick Shepard, and Giovanni Resta. A bit later, Hiroaki  Yamanouchi found
> > the next three, and recorded them in a B-file.
> >
> > Yamanouchi also found 168 more examples, for a total of 200, but was not
> > confident enough of their consecutivity to add them to the B-file;
> instead,
> > these 200 metaperfect numbers are listed in their own file.
> >
> > In the last hour, I found, essentially by hand, an example that
> Yamanouchi
> > missed: 930 284 109 364 500, which would fit between Yamanouchi's entries
> > 65 and 66. I'm not sure exactly what to do. Perhaps we should change the
> > text of the entry so that this file is labeled "Other examples, not
> > necessarily consecutive", and add my new discovery to it?
> >
> > I would also appreciate it if somebody could verify the validity of my
> new
> > example. Its factorization is 2^2*3^4*5^3*11^2*17*19*73*83*97, and I
> claim
> > it is "metaperfect" with order 14. (Because it is not divisible by 7, it
> > has a "partner" exactly 7 times bigger, which is also metaperfect, but of
> > order 18. This one is also not in Yamanouchi's list.)
> >
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> >
>
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