[seqfan] Re: A098550

[Benoît Jubin]

I just added these comments to A098548. It still needs an upper bound
(better than the doubly exponential bound I gave a few days ago).
Benoît

On Wed, Nov 26, 2014 at 9:14 PM, Neil Sloane <njasloane at gmail.com> wrote:
> 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
>> >
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