Hi, Gene.

Thanks for the answer.  (I hope to see the proof sometime.  I wonder if there's some use for quaternions here...)

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The number, r4(n), of integer solutions (w,x,y,z) of

  w^2 + x^2 + y^2 + z^2 = n, for positive integer n, is

8 times the sum of the divisors of n which are not divisible by 4. 
Expressed another way,

r4(n)/8 is multiplicative, and

r4(p^n)/8 = 1 + p + p^2 + ... + p^n for p an odd prime,

r4(2^n)/8 = 1 + 2 for n>0.
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This formula, which  I'm seeing for the first time, is amazingly interesting!

--Dan