[seqfan] Re: Another characterization of A244031?
Neil Sloane
njasloane at gmail.com
Sat May 19 15:40:42 CEST 2018
I've added "n>1" as DJS suggests.
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 Fri, May 18, 2018 at 8:03 AM, David Seal <david.j.seal at gwynmop.com>
wrote:
> Aren't the two definitions different for n=1?
>
> * It meets the 'original definition' because there is no prime strictly
> between 1 and 2, so x^2 + y^2 does not represent any such prime.
>
> * It does not meet the proposed rephrasing, because 1 + 1^2 is prime and 1
> <= 1^2 <= 1.
>
> As 1 is not in the sequence, I'd suggest that if the 'original definition'
> is the better one, it does need at least a slight modification to exclude
> 1, such as starting "Positive numbers n > 1 such that ...".
>
> Regards,
>
> David
>
>
> > On 08 May 2018 at 16:45 Neil Sloane <njasloane at gmail.com> wrote:
> >
> >
> > Concerning A244031, about which Maximilan said
> >
> > `The current definition is indeed a bit "obfuscated" (i.e. useless
> > complicated), and I'd be in favour of the proposed rephrasing.`
> >
> > let me say that I prefer the original definition, which is
> >
> > Positive numbers n such that the quadratic form x^2+n*y^2 does not
> > represent a prime strictly between n and 2n.
> >
> > and is based on the paper by my friends Bill Jagy and Kap Kaplansky.
> > I don't agree that it is "useless complicated".
> >
> > 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 Tue, May 8, 2018 at 9:52 AM, Marc LeBrun <mlb at well.com> wrote:
> >
> > > If you haven't already, I suggest that if you change the definition you
> > > still retain a comment giving the alternate form (eg to help folks
> > > searching on "quadratic form").
> > >
> > > > On May 7, 2018, at 7:36 PM, David Wilson <davidwwilson at comcast.net>
> > > wrote:
> > > >
> > > > Aha.
> > > >
> > > > I generated the sequence independently using my simpler definition,
> and
> > > looking up the terms yielded A244031.
> > > > When I saw that A244031 was NJAS's sequence, I assumed the quadratic
> > > form characterization was a necessary part of the definition.
> > > > But as MFH show, the quadratic form characterization trivially
> devolves
> > > to my simpler characterization.
> > > >
> > > > I assume this is also the case for A244029 and A244030 as well (which
> > > appear to be the prime and composite elements, respectively, of
> A244031).
> > > >
> > > > Conjecturally, all these sequences are finite.
> > > >
> > > >> -----Original Message-----
> > > >> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> M. F.
> > > >> Hasler
> > > >> Sent: Monday, May 07, 2018 12:32 PM
> > > >> To: Sequence Fanatics Discussion list
> > > >> Subject: [seqfan] Re: Another characterization of A244031?
> > > >>
> > > >> On Sat, May 5, 2018 at 8:47 AM, David Wilson wrote:
> > > >>
> > > >>> n such that 1 <= k^2 <= n => n + k^2 is composite.
> > > >>>
> > > >>
> > > >> It's easy to see that this is completely equivalent, because y has
> to
> > > be equal
> > > >> to 1 in order to have
> > > >> x^2 + n y^2 strictly between n and 2n (and x^2 is never prime), so
> for
> > > the
> > > >> considered purpose,
> > > >> x^2 + n y^2 is equivalent to n + x^2.
> > > >>
> > > >> The current definition is indeed a bit "obfuscated" (i.e. useless
> > > complicated),
> > > >> and I'd be in favour of the proposed rephrasing.
> > > >>
> > > >> - Maximilian
> > > >>
> > > >> --
> > > >> Seqfan Mailing list - http://list.seqfan.eu/
> > > >
> > > >
> > > > --
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> > >
> > >
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> > >
> >
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