[abetusk]

I asked OpenAI.

Take an $n$, chosen from $[N,2N]$. Take it's prime factorization $n = \prod_{j=1}^{k} q_j^{a_j}$. Take the logarithm $\log(n) = \sum_{j=1}^{k} a_j \log(q_j)$.

Divide by $\log(n)$ to get the sum equal to $1$ and then define a weight term $w _ j = a_j \log(q_j)/\log(n)$.

Think of $w_j$ as "probabilities". We can define an entropy of sorts as $H_{factor}(n) = - \sum_j w_j \log(w_j)$.

The mean entropy is, apparently:

$$ E_{n \in [N,2N]}[ H_{factor}(n) ] = E_{n\in[N,2N] [ - \sum_j w_j(n) \log(w_j(n)) ] $$

Heuristics (such as Poisson-Dirichlet) suggest this converges to 1 as $N \to \infty$.

OpenAI tells me that the reason this might be interesting is that it's giving information on whether a typical integer is built from one, or a few, dominant prime(s) or many smaller ones. A mean entropy of 1 is saying (apparently) that there is a dominant prime factor but not an overwhelming one. (I guess) a mean to 0 means dominant prime, mean to infinity means many small factors (?) and oscillations mean no stable structure.