Is there a term for this number theoretic object?

Question

Given integer $N$ let $N=\prod_{i=1}^{\omega(N)}p_i^{a_i}$ be its unique prime factorization where $p_i$ are distinct primes (a total of $\omega(N)$ of them).

Is there a name for $\sum_{i=1}^{\omega(N)}{a_i}$?

Answer 1

$$f(n) = \sum_{p^k \| n} k $$

Also note $f(n) = \sum_{p^k | n} 1$ so that $$\sum_{n=1}^\infty f(n) n^{-s} = \zeta(s) \sum_{p^k} p^{-sk}$$

From which we can deduce the asymptotic and even an explicit formula for $\sum_{n \le x} f(n)$

Answer 2

$\Omega(n)$. See OEIS sequence A001222.