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

[Allan Wechsler]

Thank you, Michel, I have double-checked and my value for metasigma(3^4)
was wrong. That means the two new examples I reported were incorrect. Sorry
for the false alarm, everybody -- back to the mines for me!

On Fri, Mar 29, 2024 at 4:38 AM Michel Marcus <michel.marcus183 at gmail.com>
wrote:

> 2^2*3^4*5^3*11^2*17*19*73*83*97 = 930284109364500 does not seem to work for
> me ??
>
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> Le ven. 29 mars 2024 à 06:43, Allan Wechsler <acwacw at gmail.com> a écrit :
>
> > 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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