[seqfan] Re: 448 ways

[Frank Adams-Watters]

A good average for A002635 (number of ways to represent n 
as the sum of 4 squares) would enable us to improve this lower 
bound considerably. This would also be interesting in and of itself.

(One has to look for an average, no a lower bound, as there are 
infinitely many n with A002635(n) equal to any given k > 0.)

There is no such average in A002635 itself; I did not look at any 
of the references.

Franklin T. Adams-Watters


-----Original Message-----
From: israel <israel at math.ubc.ca>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Wed, Nov 15, 2017 8:52 pm
Subject: [seqfan] 448 ways

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--Seqfan Mailing list - http://list.seqfan.eu/