[seqfan] Re: 72 is the only pronic and powerful?

Tanya Khovanova mathoflove-seqfan at yahoo.com
Fri Nov 30 18:26:14 CET 2012


I think it is interesting that numbers are both powerful and pronic. And the fact that two people replied to me means that the discussion caught some interest.



________________________________
 From: Charles Greathouse <charles.greathouse at case.edu>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu> 
Sent: Friday, November 30, 2012 12:06 PM
Subject: [seqfan] Re: 72 is the only pronic and powerful?
 
It's readily derived from A060355, and I'm not sure that it needs to be
submitted separately. What do you think, Tanya?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Fri, Nov 30, 2012 at 11:58 AM, Tanya Khovanova <
mathoflove-seqfan at yahoo.com> wrote:

> Superb!
>
> Can someone submit?
>
> Tanya
>
>
>
> ________________________________
>  From: Charles Greathouse <charles.greathouse at case.edu>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Sent: Friday, November 30, 2012 11:48 AM
> Subject: [seqfan] Re: 72 is the only pronic and powerful?
>
> The next few powerful pronic numbers are 83232, 456300, 96049800,
> 148048056, 55330565400, 110841386112, ... and there are infinitely many.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Fri, Nov 30, 2012 at 11:30 AM, Tanya Khovanova <
> mathoflove-seqfan at yahoo.com> wrote:
>
> > Dear SeqFans,
> >
> >
> > I just received the following email from Paul Wright:
> >
> > "Just thought I'd share a property that I didn't see listed on
> > NumberGossip.com. 72 is the ONLY number that is both pronic and powerful,
> > since 8 and 9 are the only consecutive non-trivial powers, as
> demonstrated
> > by Mihăilescu's 2002 proof of Catalan's Conjecture.  NumberGossip lists
> 72
> > as both pronic and powerful, but doesn't mention that it is, in fact, the
> > only number which satisfies both conditions."
> > There is a flaw in this reasoning. Suppose n and n+1 are both powerful,
> > but not powers. Then their product will be powerful and pronic (pronic
> > means the product of two consecutive numbers). I wonder if the proof of
> > Catalan's conjecture extends to powerful numbers, not just powers.
> >
> > Does anyone know anything about this?
> >
> > Tanya
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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>
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