Functional Diophantine Equation

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

Evaluate the number of triples of positive integers (a,b,n)(a,b,n) satisfying the equation f(n)=a3+b3.f(n) = a^3 + b^3.

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