Count numbers of a power number

Question

I have a power number $((a^b)^c)^d$. I want to know how many number will be produced if the terms $a,b,c,d$ will run from $1$ to $n$. Can anyone help me a general formula to calculate myself, or a result with $n$, say, $16$?

Thanks.

Answer 1

If you are simply talking about how many numbers can be produced, then there are $n$ possible values for each of $a$, $b$, $c$, and $d$. In that case there are $n^4$ possibilities but this answer does not consider distinct numbers and will count duplicate numbers more than once.

If you want distinct numbers, then we can find combinations with $b$, $c$, and $d$: $\begin{pmatrix} n\\3\end{pmatrix}$ for a total of $n\cdot\begin{pmatrix} n\\3\end{pmatrix}$ possibilities.

But this is still a limited answer as it does not consider factors of the combinations nor the fact $1^n=1$ for any $n$.