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

[Allan Wechsler]

It's been a minute, as the kids say, since I posted my spurious
"metaperfect" finds. The relevant sequence is https://oeis.org/A068978.

Since then, I wrote some software that searches for these guys to a
pre-programmed "depth", which has nothing to do with the size of the
candidates, but rather limits how "bad" intermediate candidates are allowed
to get while looking for actual examples. Details are available on request.

For a sufficient "depth", I find all 200 of Hiroaki Yamanouchi's examples,
and no new examples in the same range. This increases my confidence
Yamanouchi's "a-file" is exhaustive, but I'm certainly not 100% convinced.

The program also found 630 larger examples; the largest, of index 49, is

490 34626 57641 41000 48697 47918 18211 92775 78976 62018 64216 24891 98161
53438 01472 23286 45327 77226 45120 00000.

This is right around half of one percent of a googol. Like most numbers of
this general sort, it is easy to factor; its highest prime factor is 310433.

If anybody is interested in this sequence, I will be happy to share my
data, which is an 830-line file.

Are there other lists of large examples beyond Yamanouchi's list of 200? Is
there any interest in compiling these lists and posting it as an attachment
to the entry for A068978. Again, I emphasize that my list is almost
certainly *not* consecutive beyond the Yamanouchi Zone, but I thought there
might be interest in an archive of known examples, to which new discoveries
could be added.

On Fri, Mar 29, 2024 at 8:24 PM <hv at crypt.org> wrote:

> I think this is a lot more context-dependent than to justify a blanket ban.
>
> I guess the distinction I'd draw is how "b-file-like" the a-file is:
> the only times I've ever used a-files is because the data doesn't _quite_
> fit the b-file format, usually because there's an unknown value.
> If someone has new information to add to such a file, I'd expect them do so
> in a way that preserves the existing information, and I would hope that
> editors verify such changes the same as they would for primary data lines
> of a sequence, or for an updated b-file.
>
> In that context I'd see no problem with an updated a-file replacing the
> previous one. Similarly if someone discovered errors in an existing
> a-file.
>
> Hugo
>
> "M. F. Hasler" <oeis at hasler.fr> wrote:
> :I don't think that we should edit contents of a-files submitted by someone
> :else. We wouldn't take a PDF or image file uploaded in the very same way
> :and modify it to add information, so why only for .txt files?
> :
> :I think there are many example of a similar situation where the new
> :contributor would (or has) rather upload(ed) a new file, in addition to
> the
> :existing one. (With name and date.) (And in cases where just one new value
> :is found, there's often a comment like "(The list lacks ... and maybe
> other
> :values. - ~~~~)" right next to the link.
> :
> :Especially if not only lines are added but indexing is changed or
> :suppressed. As others mentioned, it would make it impossible to refer to
> :"Yamanouchi's fourth value" or the like, and it might be desirable and
> :useful to do this.
> :
> :
> :- Maximilian
> :
> :
> :On Thu, Mar 28, 2024, 22:24 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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