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 \[f(n) = a^3 + b^3.\]