When is the sum of divisors a prime?

Question

Can we efficiently figure out when the sum of divisors of a number can be a prime?

I realized that this can be possible only when the number is expressible as a power of only one prime, e.g. $n = p^\alpha$. Now, the sum of divisors is $ 1+p+p^2+p^3+ \ldots + p^\alpha$. Now the problem is to figure out when this summation could be prime. How do we go about it?

Answer 1

This is sequence A023194 of OEIS ($\sigma_1$ is the divisor function).

Not much seems known except that all solutions except $n=2$ may be written as $\ n=p^{2m}$ and have a prime number of divisors (i.e. $2m+1$ is prime).

Sorry if this doesn't help,