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

[Ami Eldar]

Thank you for your replies.
I have added comments in these sequences, as a conjecture in A230655 and as
a fact in A071383.

On Wed, Sep 11, 2019 at 8:30 PM <israel at math.ubc.ca> wrote:

> For a number n whose prime factors are == 1 (mod 4), each prime factor p
> can be written uniquely as x^2+y^2 with 0<x<y, corresponding to the
> Gaussian prime factorization p = (x+iy)(x-iy). The number of positive
> integer solutions of n = x^2+y^2 should then be the same as the number of
> divisors of n. Thus (unless n itself is a square) the number of points of
> Z^2 on the circle of radius sqrt(n) is 4 times the number of divisors of n.
>
> Cheers,
> Robert
>
> On Sep 11 2019, Ami Eldar 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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> >
> >
>
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