Limiting behavior of this sequence of integers For $n\in \Bbb N \setminus \{1\}$ we have $n={p_1}^{k_1} \cdots {p_m}^{k_m}$ uniqely, by the fundamental theorem of arithmetic.

We can now define sequence $\alpha(n)$ to be $\alpha(n)=\sum_{r=1}^m k_r$ and take as a convention that $\alpha(1)=0$.

Then we can define sequence $\beta(m)=\sum_{w=1}^m \alpha(w)$.


  Does there exist real numbers $q_1$ and $q_2$ such that we have $$\lim_{d \to \infty} \frac {\beta(d)}{{q_1} \cdot{d}^{q_2}}=1$$
Let $$\Omega(n) = \sum_{p^k \| n} k$$ Then $$ \beta(N)= \sum_{n \le N} \Omega(n) = \Omega(N!)= \sum_{p^k \le N}  \lfloor N/p^k \rfloor= N\sum_{p^k \le N}  \frac{1}{p^k}+\mathcal{O}(N) = N \log \log N+\mathcal{O}(N) $$
by Mertens 2nd theorem
$\beta(n)$ is OEIS A022559. An asymptotic expression is given there by Charles R. Greathouse IV:
$$\beta(n)=n\log\log n+1.0346n+o(n)$$
So the $q_1$ and $q_2$ as constants in the proposed limit do not exist, since $\beta(n)$ exhibits quasilinear rather than polynomial growth.