In number theory, is there the idea of a "metaprime" -- a number with a prime number of factors?

Question

This could be the definition or perhaps the sum of its factors being prime. There must be many numbers with a prime number of factors, in fact every prime is a metaprime by this definition but I don't know about the sum of all factors or prime factors. 9 would be a metaprime where all (positive) factors add up to 13. I am wondering if this is even an area of study -- I can't find anything when I google it.

Answer 1

The class of numbers with prime number of factors is easily described:

Suppose $\tau (n)$ is the number of factors of $n$. We know that $\tau$ is a multiplicative function and that $\tau (n) = 1 \iff n = 1$. That means, that $\tau (n)$ can be prime only in case, when n is a degree of a prime. For prime p and natural m, we have $\tau(p^m) = m+1$. Thus all numbers with prime number of factors have the form $p^{q-1}$ where both $p$ and $q$ are primes.

The class of numbers with prime sum of factors has a much more complex structure, but can be described as well:

Suppose $\sigma (n)$ is the sum of factors of n. We know that $\sigma$ is a multiplicative function and that $\sigma (n) = 1 \iff n = 1$. That means, that $\sigma (n)$ can be prime only in case, when n is a degree of a prime. For prime $p$ and natural $m$, we have $\sigma(p^m) = \sum_{k = 0}^{m} p^k$ . Thus all numbers with prime sum of factors have the form $p^m$, where both $p$ and $R_{m+1}^{(p)}$ are primes.

The class of numbers with prime number of factors seems somewhat trivial, but the class of numbers with prime sum of factors is strongly connected with the theory of repunit primes, and that is a developing "non-elementary" branch of "elementary number theory" full of various unsolved problems: https://en.m.wikipedia.org/wiki/Repunit