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Italian functions.

Source: Italy National Olympiad 2020 P5

September 30, 2020
functionnumber theoryprime numbers

Problem Statement

Le SS be the set of positive integers greater than or equal to 22. A function f:S→Sf: S\rightarrow S is italian if ff satifies all the following three conditions: 1) ff is surjective 2) ff is increasing in the prime numbers(that is, if p1<p2p_1<p_2 are prime numbers, then f(p1)<f(p2)f(p_1)<f(p_2)) 3) For every n∈Sn\in S the number f(n)f(n) is the product of f(p)f(p), where pp varies among all the primes which divide nn (For instance, f(360)=f(23⋅32⋅5)=f(2)⋅f(3)⋅f(5)f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)). Determine the maximum and the minimum possible value of f(2020)f(2020), when ff varies among all italian functions.
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