Question 29
For every positive integer $n$ with prime factorization $n=\prod_{i=1}^{k} p_{i}^{\alpha_{i}}$, define
$$
\mho(n)=\sum_{i: p_{i}\gt 10^{100}} \alpha_{i} .
$$
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
\mho(f(a)-f(b)) \leqslant \mho(a-b) \quad \text { for all integers } a \text { and } b \text { with } a\gt b .
$$
$$
\mho(n)=\sum_{i: p_{i}\gt 10^{100}} \alpha_{i} .
$$
That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity.
Find all strictly increasing functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$$
\mho(f(a)-f(b)) \leqslant \mho(a-b) \quad \text { for all integers } a \text { and } b \text { with } a\gt b .
$$