[seqfan] 448 ways

[israel at math.ubc.ca]

A295159 (recently submitted by Robert Price) is titled "Smallest number 
with exactly n representations as a sum of five nonnegative squares." Of 
course this raises the question, does such a number always exist?

It appears to me that although such numbers exist for n <= 447, there may 
not be such a number in the case n=448. I have determined that there is no 
such number <= 25000, and from looking at A000174 it seems unlikely that 
any number x > 25000 would have A000174(x) as small as 448. Can anybody 
prove my conjecture?

One possible route to proving the conjecture would involve good lower 
bounds for A000174(x). One simple lower bound is floor(sqrt(x) - 
sqrt(x/2)), from the following argument: if y is any integer in the 
interval (sqrt(x/2), sqrt(x)], we can represent x as the sum of 5 squares 
of which one is y^2 (since x - y^2 is the sum of 4 squares), and since y^2 
> x/2 the others are all < y^2, making these all distinct representations. 
To make that lower bound 448 would require x >= 2339578, and that's too big 
for me to search. But maybe better lower bounds are known.

Cheers,
Robert Israel