Think Logically not mathematically [part-32]

Algebra Level 4

\[ \begin{cases} f(mn) & = & f(m)f(n) \\ f(m) & < & f(n) & \text{if} \ \ m<n \\ f(2)& = & 2 \end{cases}\]

Suppose \(f: \mathbb {N \rightarrow N} \) is a function satisfying the conditions above. Find the value of \(\displaystyle \sum_{k=1}^{20} f(k)\).

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