Re: question on fitting a function to an actual distribution

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Wed, Nov 26, 2025 at 04:53:30PM -0800, American Citizen wrote:
> Hello:
> 
> I have been pushing forward, using a function in gp-pari script written by
> Max Alekseyev, which finds how many sets of square triads might sum to a
> given number. For example n=194 results in [[0, 5, 13], [1, 7, 12], [3, 4,
> 13], [3, 8, 11], [5, 5, 12], [7, 8, 9]] or six triads, the number 194 is the
> smallest for 6 triads. His program works very well, and I verified it for
> the values of 1 <= n <= 1e7. (first 240 values of rel max highs)
> 
> I have a collection of 41,374 triad counts for the 1 <= count <= 41374 with
> the n value given.
> 
> This distribution seems to follow:
> 
> (1)  n =  5.32459104391076 * count ^ 1.93934231682073
> 
> which is a linear graphic when plotted on a log-log scale.

I think using the theta function definition of the number of triple lead
to a much more natural formula.
See https://www.cirm-math.fr/RepOrga/2062/Slides/Humphries.pdf

Cheers,
Bill