Question 55
Positive integer $n$ is called 'nice' if the sum of squares of its divisors is equal to $(n+3)^2$ (for example $287=7 \cdot 41$ is 'nice', since $1^2+7^2+41^2+287^2= 84100=(287+3)^2$). Prove that if for two primes $p$ and $q$ (not necessarily distinct) the number $n=pq$ is 'nice', then both $n+2$ and $2(n+1)$ are perfect squares.