Let M be the smallest positive multiple of 2012 that has 2012 divisors. Suppose M can be written as
\prod_{k=1}^{n} p_{k}^{a_{k}}
where the p_{k} 's are distinct primes and the a_{k} 's are positive integers. Find
\sum_{k=1}^{n}\left(p_{k}+a_{k}\right).