Let C(n) be the sum of the squares of the divisors. We have $2A(n)|A(n)^2-C(n)$. Assume that $2 |A(n)$. $2 \prod \frac{p^{a+1}-1}{p-1} | \prod \frac{p^{2(a+1)}-1}{p^2-1} \implies 2|\prod \frac{p^{a+1}+1}{p+1}$. Note that $v_2(\frac{p^{a+1}+1}{p+1}) \le 0$ for any $p$, contradiction. Thus $2$ does not divide $A(n)$ hence powers of primes in $n$ except $2$ are even. If there is a prime with $3|\frac{p^{2a+1}+1}{p+1}$ (iff $p \equiv 2 \mod 3$ and $a \equiv 1 \mod 3$ by LTE) then power of two can take any value. Otherwise it must be even (so $n$ is a perfect square). These clearly satify the conditions.