Find an algebraic function $f:\Bbb N\to\Bbb N$ such that
$$f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$$
and
$$f(1)=1$$
for all $x\in\Bbb N$
I allready know two things:
$f(p^k)=p$ where $p$ is prime and $k\in\Bbb N$
$f(apq)=1$ where $p$ and $q$ is two different primes and $a\in\Bbb N$
I'm looking for a specific function, not just an algorithm.
You have $$n = \prod_{d\mid n} f(d)$$ so you can use the product version of Möbius inversion to get:
$$f(n)=\prod_{d\mid n} d^{\mu(n/d)}$$
Not sure if that helps, nor if there isn't some other, simpler closed form.
For $n=p^k$ with $p$ prime, we see this is $f(p^k)=\frac{p^k}{p^{k-1}}=p$.
For $n=p^aq^b$, this is $$\frac{(p^aq^b)(p^{a-1}q^{b-1})}{(p^{a-1}q^b)(p^aq^{b-1})}=1$$
I'm guessing it is $f(p^k)=p$ and $f(n)=1$ for $n$ not a prime power.
It certainly seems to work...