[seqfan] Re: A098550

David Applegate david at research.att.com
Wed Nov 26 22:05:39 CET 2014


In Benoit Jubin's email, he noted (about A098548)

> > Empirically, a(2n+1) seems to be a multiple of 3
> > and a(2n+1)-a(2n) seems to be prime (5,11,17...)

These aren't quite always true.  Eventually we get to

a(491) = 1869
a(492) = 1870
a(493) = 1883

and 1883 is not a multiple of 3.  Continuing,

a(494) = 1884
a(495) = 1897
a(496) = 1898
a(497) = 1925

and a(497)-a(496) = 27, not a prime.

-Dave

> From seqfan-bounces at list.seqfan.eu Wed Nov 26 15:15:47 2014
> Date: Wed, 26 Nov 2014 15:14:51 -0500
> From: Neil Sloane <njasloane at gmail.com>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: A098550

> Benoit, Very interesting!

> Would you please add your comments to A098548?
> And also send in the two sequences of square-free parts (I assume that's
> what you meant?).

> Concerning A098550, I can prove that the sequence is infinite, and that for
> any prime p, there is a term divisible by p. Rather pathetic.

> At various times during the past days I thought I had proofs that (a) any
> prime p is eventually in the sequence on its own, (b) every p divides
> infinitely many terms, (c), every p^k is a term,and (d), every m is a term.
> Or even (e) there are infinitely many even terms.  However, none of these
> "proofs" have survived till the next day...

> Best regards
> Neil

> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com

> On Wed, Nov 26, 2014 at 12:39 PM, Benoît Jubin <benoit.jubin at gmail.com>
> wrote:

> > Dear Frank and seqfans,
> >
> > This is interesting (that A098550(n)/n has a discrete set of adherence
> > values). Do you have approximate values for the first few?
> > Have you looked at the probably simpler sequence A098548?
> > For it, one can prove that:
> > a(n) is even if and only if n is even
> > a(2n) = a(2n-1)+1
> > a(2n+1)-a(2n) is at least 5
> > and in particular a(n)>3n for n large enough, but I cannot prove that
> > a(n)<Kn for some K. Empirically, a(2n+1) seems to be a multiple of 3
> > and a(2n+1)-a(2n) seems to be prime (5,11,17...) and a(n)/n seems to
> > have a limit close to 4.
> >
> > I think it would be worth adding the squarefree parts of A098548 and
> > A098550.
> >
> > Benoît
> >
> > On Fri, Nov 21, 2014 at 6:07 PM, Frank Adams-Watters
> > <franktaw at netscape.net> wrote:
> > > This sequence has what at first seems to be at most a marginally
> > interesting
> > > graph: several straight lines. But when we look at a(n)/n:
> > >
> > >
> > https://oeis.org/plot2a?name1=A098550&name2=A000027&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true
> > >
> > > it gets more interesting. The lines do not have integral slope, as one
> > would
> > > expect them to have. Any insights into what what is going on here?
> > >
> > > Franklin T. Adams-Watters
> > >
> > > _______________________________________________
> > >
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >

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