Question 152
For a positive integer $m$, let $d(m)$ denote the number of positive divisors of $m$ ($1$ and $m$ included). Let $k$ be a positive integer. Show that there exist infinitely many positive integers $n$ such that $n$ has exactly $k$ distinct prime divisors and such that for all positive integers $a, b$ with $n=a+b$, $d(n)$ does not divide $d\left(a^{2}+b^{2}\right)$.