# When is the sum of divisors a prime?

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?

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I don't think you'll have much luck here, with $p=2$ you get the Mersenne primes for example –  Cocopuffs Aug 25 '12 at 7:01
And when $\alpha=2$ you're asking for primes of the form $p^2+p+1$, another notorious open problem. –  Gerry Myerson Aug 26 '12 at 6:43
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).