[seqfan] Re: Your recent Flavius Josephus's sieve related sequence. (and to Olivier: Problem with SeqFan-archives?)

David Wilson davidwwilson at comcast.net
Thu Nov 24 16:52:52 CET 2016

A193764 and A278484 are both O(n^2), but they don't grow at exactly same rate.

For A193764 we have:
%F For n > 6, except n = 13, a(n) = n^2 + 4 - Floor((n+2)^2/5).
so that

A193764(n) ~ 0.8 n^2.

For A278484 we have
%F a(n) = A000960(n + 1) - 1. (Note: This is provable, so I removed the NJAS comment asking if it is a conjecture).
and for A000960 we have
%F Let F(n) = number of terms <= n. Andersson, improving results of Brun, shows that F(n) = 2 sqrt(n/Pi) + O(n^(1/6)). Hence a(n) grows like Pi n^2 / 4.
so

A278484(n) ~ pi/4 n^2 ~=  0.78539 n^2.

> -----Original Message-----
> From: Antti Karttunen [mailto:antti.karttunen at gmail.com]
> Sent: Thursday, November 24, 2016 9:46 AM
> To: David Wilson
> Cc: Charles Greathouse
> Subject: Re: Your recent Flavius Josephus's sieve related sequence. (and to
> Olivier: Problem with SeqFan-archives?)
>
> BTW,
>
> https://oeis.org/A193764
> grows with almost the same pace as:
> https://oeis.org/A278484
>
> Maybe this is obvious.
>
> Best,
>
> Antti
>
>
> On Thu, Nov 24, 2016 at 1:24 AM, Antti Karttunen
> <antti.karttunen at gmail.com> wrote:
> > Now pushed everything for approval (only 22 edits, whoah..., many
> > slightly different variants, because I will use them one day for other
> > sequences.)
> >
> > If you happen to find a more direct way (that just searching with a
> > loop) to compute these two, then please edit. I surmise there might be
> > even a closed form for at least the other (guessing just on the
> > grounds of the fact that this Flavius-sieve seems to be the most
> > regular of all alternative sieve-variants, lucky, ludic, etc.):
> >
> > https://oeis.org/draft/A278528
> >
> > https://oeis.org/draft/A278529
> >
> >
> > Best regards,
> >
> > Antti
> >
> >
> > On Wed, Nov 23, 2016 at 4:22 PM, Antti Karttunen
> > <antti.karttunen at gmail.com> wrote:
> >> FYI:
> >>
> >> https://oeis.org/draft/A278484
> >>
> >> I guess this is clear: a(n) = 2*A073359(n) as observed by Arie
> >> Groeneveld ?
> >>
> >>
> >> https://oeis.org/draft/A278482
> >>
> >> https://oeis.org/draft/A278492
> >>
> >> etc.
> >>
> >> Please feel free to make further edits (but I'm still editing these
> >> for an hour or two, so beware of any transaction-clashes...)
> >>
> >> Actually, I want to create https://oeis.org/A255127 analogue for this
> >> (with similar indexing, starting with a(2) = 2), and a couple of
> >> related sequences, like the index-sequences for the rows and for the
> >> columns, for my sub-collection of sieves. Doing it tonight.
> >>
> >>
> >> Best regards,
> >>
> >> Antti
> >>
> >> PS. I would like to make a reference to the appropriate posting in
> >> SeqFan-list archives, but currently the archive is not up to date:
> >>
> >> http://list.seqfan.eu/pipermail/seqfan/2016-November/date.html
> >>
> >> CC: to Olivier, maybe he can investigate?