OSN 2007 Problem 2

For every positive integer $n$ , $b(n)$ denote the number of positive divisors of $n$ and $p(n)$ denote the sum of all positive divisors of $n$ . For example, $b(14)=4$ and $p(14)=24$ . Let $k$ be a positive integer greater than $1$ .

(a) Prove that there are infinitely many positive integers $n$ which satisfy $b(n)=k^2-k+1$ .

(b) Prove that there are finitely many positive integers $n$ which satisfy $p(n)=k^2-k+1$ .