This question is a follow up to my comment to [Sum of the reciprocal of perfect numbers](https://mathoverflow.net/questions/99227/sum-of-the-reciprocal-of-perfect-numbers). I would like to know which results have been published about the possible irrationality of the sum of reciprocals of perfect numbers. For example, do we know a lower bound for the denominator of this number provided it is rational?  

Thanks in advance.



Any results towards the irrationality of the sum of reciprocals of perfect numbers?
As Stanley Yao Xiao commented, a definite answer to this question would be equivalent to solving an open problem. If we assume two reasonable conjectures, however, then the sum $\sigma$ of the reciprocals of the perfect numbers is irrational:


* The Lenstra-Pomerance-Wagstaff conjecture that there are asymptotically $k \log{\log{N}}$ Mersenne primes below $N$, where $k$ is some real constant.
* The conjecture that no odd perfect numbers exist.


Then we can find infinitely many sufficiently good rational approximations (of the form $|\sigma - \frac{p}{q}| < \frac{1}{q^{\alpha}}$, where $\alpha > 1$ is a function of $k$), which implies that $\sigma$ is irrational.


