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

[Ami Eldar]

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