[seqfan] Re: A230655 & A071383 and (3, 1)- and (4, 1)-highly composite numbers.

[Allan Wechsler]

This isn't the first time I've heard about correlations between the number
of representations of a given integer by a particular quadratic form, and
the number of divisors of some particular form of that integer (or some
related integer). There are lots of theorems of that sort.

Representations by a quadratic form, of course, correspond to integer
points on a conic section (for example, a circle). The most basic example
is that the lattice points on the positive branch of a hyperbola xy=n
correspond to all the divisors of n.

The theory is rich and beautiful. Gauss ran the first exploratory trenches
(in his "Disquisitiones Arithmeticae"), and of late it has ramified into
the theory of modular forms and modular groups. I know very little of it.

On Wed, Sep 11, 2019 at 12:04 PM Neil Sloane <njasloane at gmail.com> wrote:

> Amiram,  Those seem to be remarkable coincidences!
>
> Since they agree for so many terms, I think you should add comments to both
> sequences saying something like "It appears that ..."
>
> Very interesting!
>
>
> 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, Sep 11, 2019 at 8:05 AM Ami Eldar <amiram.eldar at gmail.com> wrote:
>
> > Hello,
> >
> > There are variations of highly composite numbers (numbers with a record
> > number of divisors, A002182) in literature in which the divisors are
> > restricted to have prime factors only of the form a*k+b (a and b
> coprime, k
> > = 0, 1, 2, ... ). These numbers may be called (a,b)-highly composite
> > numbers (the usual highly composite numbers are then (1, 0)-highly
> > composites).
> >
> > I have calculated the sequences of (3,1)- and (4,1)-highly composite
> > numbers, and apparently they are already in the OEIS, but with a
> different
> > interpretation: A230655 and A071383 which are the sequences of squared
> > radii of the circles around a point of hexagonal (A230655) or square
> > (A071383) lattice that contain record numbers of lattice points.
> >
> > I have compared terms with all available data in these two sequences (17
> in
> > A230655 and 97 in A071383) and found that they are the same.
> >
> > In both sequences, divisors are not mentioned, but in A071383 it is said
> > that all the terms are products of consecutive primes of the form 4k+1
> > starting from 5, with nonincreasing exponents, which is necessary, but
> not
> > sufficient, for the terms to be highly composite.
> >
> > Can it be proven that the two definitions are equivalent? If yes, I think
> > that it worthwhile to mention it.  In addition, in that case I can easily
> > extend the data with many more terms.
> >
> > Best,
> >
> > Amiram
> >
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