MathDB

Find all functions

f : \mathbb{N} \rightarrow \mathbb{N}

such that the following conditions are true for every pair of positive integers

(x, y)

(i)

x

and

f(x)

have the same number of positive divisors.

(ii)

: If

x \nmid y

and

y \nmid x

, then:

\gcd(f(x), f(y)) > f(\gcd(x, y))

5