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# Functional equation

Let $$f$$ be a function defined as $$f: \mathbb N \to \mathbb N$$, where $$\mathbb N$$ denotes the set of natural numbers, and $$f$$ satisfy the following conditions:

(a): $$f(m) < f(n)$$ for all $$m<n$$.
(b): $$f(2n) = f(n)+n$$ for all positive integers $$n$$.
(c): $$n$$ is a prime whenever $$f(n)$$ is a prime.

Find $$f(2001)$$.

Note by Kalpa Roy
11 months, 1 week ago

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What have you tried? What are your thoughts?

If we removed the 2nd condition, then there are a lot of possibilities, like making $$f(n)$$ the nth prime. So, how can we make use of the 2nd condition? Staff · 11 months, 1 week ago