# Functional Diophantine Equation

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 $f(n) = a^3 + b^3.$

×