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

[Neil Sloane]

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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