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

[Michel Marcus]

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.)
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
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> >
>
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