A071383, A071384, A071385

[Hugo Pfoertner]

SeqFans,

%I A071383
%S 0,1,5,25,65,325,1105,4225,5525,27625,71825,138125,160225,801125,2082925,
%N Squared radii of the circles around (0,0) that contain record numbers of lattice points

In his analysis of this sequence on May 31 2002 John Conway wrote:
<<
But let me think about this sequence f(n). It's clear that the
squared radius is a product of primes of the form 4n+1, and that
one-quarter of the number of points is the product of a factor for each
exact prime power divisor P = p^k of N = f(n).
What is this factor? Well, for
P prime, say 5, it's 2 (1 from 2+i, 1 from 2-i)
P = p^2, say 25, it's 3 (from (2+i)^2, (2+i).(2-i), (2-i)^2)
and so on, making it obvious that the factor is just k+1. This
evaluates the numbers of points corresponding to the first terms thus:
N                 # of points
1    =  1         1
5    =  5         2
25   =  5^2       3
65   =  5x13      2x2
325  =  5^2x13    3x2
1105 =  5x13x17   2x2x2
4225 =  5^2x13^2  3x3
5525 =  5^2x13x17 3x2x2
It's clear that the optimal N have the form 5^a.13^b.17^c...
with a >= b >= c >= ... . So we have to increase
log # = log(a+1) + log(b+1) + log(c+1) + ...
while keeping
log N = a.log5 + b.log13 + c.log17 + ...
as small as possible.
[snip]
So the optimal N will be those for which all the p^d are
roughly the same size P,...
>>
Be curious if there is any "pattern" in the sequence of exponents
and what "roughly the same size P" means
for the record producing combinations, I have written a little
Fortran program that tries all combinations up to 2^63, which is
the limit given by my available hardware (Digital Alphastation).
The program is available at
http://www.randomwalk.de/sequences/crlmod.f   and
http://www.randomwalk.de/sequences/icirlt.f (count hit lattice points M
ignoring John Conway's result M=4*(a+1)*(b+1)*...)

A table of the results, sorted by increasing radii is at
http://www.randomwalk.de/sequences/a071393.pdf

The first 11 columns of this table contain the exponents for the first 11
primes of the form 4*n+1, the bold printed entries in the
last 3 columns correspond to A071383, A071384 and A071385.

Could the first column in this table (exponents of 5) be a candidate
for another new sequence?

One question regarding the comment in A071384:
A071384(n)^2=A071383(n) for A071384(n)=1,5,65,...
I can "see" that 65 is probably the last integer radius producing
a record number, but proving this would be better ;-)

Hugo Pfoertne