Let $n\geq 1$ be an integer, $P$ the set of rational prime numbers. I am interested in upper bounds for $\prod_{p\in P; \ p\mid n}(1+1/p)$ in terms of n.

I would like to find explicit real numbers $a,b$ such that for any integer $n\geq 1$ it holds $\prod_{p\in P; \ p\mid n}(1+1/p)\leq a\log(n)^b$.

By the answer of GH this is possible for any $b>0$ provided $n$ is sufficiently large.

Now, I would like to find explicit real numbers $a,c$ such that for any integer $n\geq 10$ it holds $\prod_{p\in P; \ p\mid n}(1+1/p)\leq a\log(n)+c$.

What are small possible values for $c$ and $a$?