A071383, A071384, A071385

Hugo Pfoertner hugo at pfoertner.org
Tue Jun 11 22:29:48 CEST 2002


%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.
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

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

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