San Bernardino Squares [Zappa..]

[Wouter Meeussen]

hi all,

Number theory is a funny thing, so here it comes ..

Why is it that 
Product[{d|n} (n/d + d) ]

series1:
{2,9,16,100,36,1225,64,2916,600,5929,144,529984,196,18225,16384,231200,324,3
538161,400,5143824,48400,89401,576,1482250000,6760,..}
(too bad EIS is out of order right now!)

is always square except just_when n is square,
unless n= k^2 and k belongs to
series2:
{2,8,18,32,50,72,98,128,162,200,242,288,338,392,450,512,578,648,722,800,882,
968,..}


example: 
k=2 -> n=4 -> ser1[4]=100=10^2 is the first exception,
k=8 -> n=64 -> ser1[64]= 31258240000 = 176800^2 is the second exception, ..

- -----------------------
why? where did I get this?
look at the coefficient list of Product[{d|n} (x n/d + d) ] and remember that
Product[(x - m), {m,0,n-1} ]=
= x!/(x-n)! =
= x (x-1) (x-2) (x-3)..(x-n+1) =
= Sum[StirlingS1[n,m] x^m, {m,n}] 

..and you see that I tried to cook up some Stirling-oid structure for
divisors only.

Take n =12 :
you then see that it is equivalent to the sum of all possible products
(a,b,c d e f)
with of the six factors a..f chosen from :
  
a {1 ,12)
b (2 ,6 )
c (3 ,4 )
d (4 ,3 )
e (6 ,2 )
f (12,1 )
 
wouter.

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