# A four-part trilogy

**Number Theory**Level 5

Define a function \(f : \mathbb{N} \rightarrow \mathbb{Z}^{+} \cup \{0\}\) such that

\(f(1) = 0, f(p) = 1\) for \(p\) prime, and \(f(mn) = mf(n) + nf(m).\)

Let \(A\) be the sum of all \(n \lt 1,000,000\) such that \(f(n) = n.\)

Let \(B\) be the maximum value of \(\dfrac{f(n)}{n}\) for \(n \le 1000.\)

Let \(C\) be the maximum value of \(f(n)\) for \(n \le 1000.\)

Finally, let \(D = f(1000) + f(42).\)

Find \(A + 2B + C + D.\)