MathDB

For every positive integer

n

with prime factorization

n = \prod_{i = 1}^{k} p_i^{\alpha_i}

, define

\mho(n) = \sum_{i: \; p_i > 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} \to \mathbb{Z}

such that \mho(f(a) - f(b)) \le \mho(a - b) \text{for all integers } a \text{ and } b \text{ with } a > b.
Proposed by Rodrigo Sanches Angelo, Brazil

N8