[seqfan] Re: Another characterization of A244031?

[David Seal]

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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