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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 example2012-08-25
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    And when $\alpha=2$ you're asking for primes of the form $p^2+p+1$, another notorious open problem.2012-08-26

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