Important Hyperperfect Find

[Alex Violette]

Hello SeqFans,

I have come to announce that the first Hyperperfect NOT of the form 3^(k-1)*(3^k-2) has been found! It is 321210648228113097833688328004214356646685508531160153551637=3^16*129140227*256572153302107*225205396855537107298933866373.

Best,

Alex Violette

[Alex Violette]

Correction: First 2-Hyperperfect number.

[Allan Wechsler]

I confess that I have never been sure what the motivation for the hyperperfect concept is. Can you say a few words about why this is a natural thing to look at? The Wikipedia article doesn't say anything about why hyperperfect number are interesting.

Congratulations on your find!

-- Allan

--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAAQNUoT09554uWinhQGDZ6mgUDeTdb%2BWyYroo3%3DRG%2BXZG81xPg%40mail.gmail.com.

[Alex Violette]

Hey Allan,
there is a thing where 2-hyperperfect numbers do come up: Let N=Perfect Number and O=2-Hyperperfect Number, if N and O are coprime then sigma(NO)-3*NO=N. sigma(n)-3n=28 is the smallest positive value A for the equation sigma(n)-3n=A known where "large" solutions are known to occur(Other such A include A=36, A=72, A=84, and more). In fact. numbers of the form NO is how I originally stumbled upon it as I first encountered 8993898150387166739343273184118001986107194238872484299445836 a few days before my announcement which is the smallest number known not of the form 28*3^n*(3^(n+1)-2) where sigma(n)-3n=28. Few days later, I realized the hyperperfect thing which was how I found the new 2-hyperperfect number in the first place. It's still up in the air if any solution not divisible by 28 exists where sigma(n)-3n=28 though I suspect it happens eventually. 

Also, thank you. This is also the first time I've managed to find a counterexample for a conjecture in a research paper(Conjecture 2 in the paper "Generalized perfect numbers" page 7) and possibly another if I understand it correctly though I didn't know this until after my find. I wish I could contact the one author who is still alive but the email in that paper didn't work.

Regards,
Alex Violette

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CADy-sGHV7jY3zvGfW4vVjatC4wsA3YM9S0mKDtKt%2B0giYK8Oig%40mail.gmail.com.

[D. S. McNeil]

Congratulations!

There's even arguably a downstream consequence, which is pretty rare for a conjecture. One of the papers which cites Bege & Fogarasi seems to have mistaken what BF conjectured and what they proved, and in a confusingly written proof appears to rely on the conjecture you've now disproved.  The paper in question has some.. difficulties.. but I thought it was unusual enough to mention.

Doug

[Antti Karttunen]

On Fri, Jun 26, 2026 at 3:12 AM Alex Violette <aviolett...@gmail.com> wrote:

Hello SeqFans,

I have come to announce that the first Hyperperfect NOT of the form 3^(k-1)*(3^k-2) has been found! It is 321210648228113097833688328004214356646685508531160153551637=3^16*129140227*256572153302107*225205396855537107298933866373.

Great!

Could you please update the comments-section of https://oeis.org/A007593 with your find?

Best regards,

Antti

 

Best,

Alex Violette

--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAAQNUoQ%3D%2Bj7JUuztXNfD_0DzYPBTG3hy-yBTObfeaEvjjBNoDQ%40mail.gmail.com.

[Alex Violette]

Hey Antti,
Sorry for being a bit late on this but I added it in earlier this week. Also, I did try to look for another one since my discovery but I did not find another(it's very hard to even get an actual example to begin with).

Regards,
Alex Violette

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAB%2B0_%3DnRndaxjgXt0pSvD6r%3D167SagoprDMd-ZZsp9o0Pc5wHw%40mail.gmail.com.