Let $m=p_1\ldots p_k$ be the prime factorization of some positive integer $m$ and $k\geq 2$.

Let $d_1,\ldots,d_{\tau(m)}$ be all divisors of $m$, where $\tau(m)$ counts the number of divisors of $m$.

For a given $k$, I am trying to find the minimal number of different values in the sequence $(\mu(d_i)\varphi(d_i))_{i=\overline{{1,\tau(m)}}}$, where $\varphi$ and $\mu$ are Euler's totient and Mobius functions, respectively.

It seems very likely to me that the value is equal to $2 ({{k-2}\choose {3}}+{{k}\choose {2}}+1)$. Moreover, is it possible to find $p_1,\ldots,p_k$ such that the number of different values of the sequence is equal to some given $l$, where $l$ takes the values between lower and uper bound?

Thank you in advance for any suggestion.