From the definition I have deduced $d(n)$ is multiplicative by letting $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where
$$d(p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k})= (a_1+1)\cdots(a_k+1)=d(p_1^{a_1})\cdot d(p_2^{a_2})\cdots d(p_k^{a_k})$$
as we know the divisors of $n$ are $d=p_1^{b_1}p_2^{b_2}\cdots p_k^{b_k}$ where each $b_k$ gives a distinct divisor in the form of $(a_k+1)$. However, I am unsure how to find the given integers, $n$, for the two points below;
- $d(n)$ is odd
- $d(n)= p_0,$ where $p_0$ is a fixed prime.