Is the b-file for "imperfect numbers" (A127724) known to be correct?

[Allan Wechsler]

I spent a few months in 2019 and 2020 obsessing about "imperfect numbers", https://oeis.org/A127724 . I found a few new ones, and my catalog has somewhere above 1600 of them. I didn't upload my database to OEIS because I had no clue whether the entries were consecutive; that is, I thought there might very well be other examples between the ones I knew.

Last fall Max Alexeyev uploaded as a b-file a magnificent collection of 1714 examples. This is great work, but I am curious to know if this list is known to be exhaustive as far as it goes. That is, is #1714 on the list definitely the 1714th imperfect number, or might there be gaps in the list?

Compare with https://oeis.org/A007691 , the classic multiperfect numbers, of which more than 5000 are known, but the b-file lists only 1600 of them, presumably because that's as far as we can go before we start worrying that there might be unknown examples in the interstices.

In both cases it would be good to have a reference, or at least an assertion, of the exhaustiveness of the list given. A complete list of known examples is also good to have, but not as a b-file unless proved exhaustive. I hope I've understood OEIS policy properly here.

-- Allan

[Max Alekseyev]

Hi Allan,

No, there are no gaps in my list (modulo bugs in my software/hardware, of course).

I would not pose it as a b-file otherwise. In general and by default we should assume that b-files in the OEIS have no gaps. There are just a few clearly articulated exceptions and sequence A127724 is not among them.

PS. As we speak, I recalled that I've extended my computations to 10^140 bound and just uploaded an updated b-file with 2133 terms.

Regards,

Max

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[Allan Wechsler]

Thank you, and great work!

-- Allan

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[sven-h...@gmx.de]

Hello,

concerning A007691 (multiperfect numbers) there is some information on Achim Flammenkamps Page about multiperfect numbers, which is linked at A007691. He proved that there are no gaps for MPNs up to e**350. He uses logarithmic scale with e (ln) on his pages sometimes, so e**350 would be about 150 decimal digits and about the first 700 MPNs. But I did not look into his small paper in DVI format he mentions  (linked on his page ) so it could be even further.

Sven

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