The number N is a sum of 4 different numbers, each being a square of one of the 4 smallest divisors of N (e.g. N=36 does not qualify since 1^2+2^2+3^2+4^2

sums up to 30, not 36.)

Provide a full list of
similar numbers or show that none exist.

Yeah, that's about it, as I found too. Here is where I got on the proof.

(not far)

N must be even, else only odd divisors squared (odd) would add to even. So 1st 2 divisors are 1 and 2. The last two, a and b have a^2 + b^2 = N-5, which is odd. So only one of the squares is odd, and this means only one of a and b are odd. Calling d the even one then d might = 2e with e prime and after trying 2(3) we see 2(5) works. But why must e be prime is where I am stuck, Cheers

*Edited on ***July 8, 2018, 4:08 pm**