Functional Diophantine Equation

Number Theory Level 5

Let $f : \mathbb{N} \to \mathbb{N}$ be a strictly increasing function such that $f(2) = 8$ and $f(ab) = f(a) \cdot f(b)$ for $\gcd(a, b) = 1$ .

Evaluate the number of triples of positive integers $(a,b,n)$ satisfying the equation $f(n) = a^3 + b^3.$