# [seqfan] Re: A098550

Neil Sloane njasloane at gmail.com
Wed Nov 26 21:14:51 CET 2014

```Benoit, Very interesting!

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.
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:
> >
> >
> >
> > 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?
> >
> >
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> >
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