Is sum of all divisors each multiplied by it's respective totient for a particular number is multiplicative when two numbers are coprime?

Question

Is the formula for nth term of sequence http://oeis.org/A057660 is multiplicative when the numbers are coprime ? If yes how to prove it? And what is the answer when they are not coprime.

Answer 1

Your function is multiplicative.

First look at the function $g(n)=n\phi(n)$. This function is multiplicative, using the fact that $\phi$ is multiplicative: For relatively prime $m,n$ we have $g(mn)=mn\phi(mn)=mn\phi(m)\phi(n)=g(m)g(n)$.

But then using the result if a function $f(n)$ is multiplicative, then so is $\sum_{d\mid n} f(d)$ (see for instance https://proofwiki.org/wiki/Sum_Over_Divisors_of_Multiplicative_Function), we get that your function, which is equal to $\sum_{d\mid n} g(d)$, is also multiplicative.