It is well known that the sum of the reciprocal of prime numbers is $+\infty$. This proves that there infinitely many prime numbers. On the other hand it is also known that the series of the reciprocal of twin prime numbers converges, so nothing can be said about the finiteness of the set of twin prime numbers. This is indeed an open problem. A similar open problem is the existence of infinitely many perfect numbers. So this is the question:


let $\mathcal{Pe}$ be the set of perfect numbers. What is known about the series
$$
\sum\_{n \in \mathcal{Pe}} \frac{1}{n} 
$$


This is just curiosity, and could be a trivial question. Note that if there are no odd perfect numbers (as it seems to be the case), the series converges. Indeed any $n$ perfect is of the form $2^{p-1}(2^p-1)$.



Sum of the reciprocal of perfect numbers
A Google search finds papers [estimating the sum of the reciprocals of amicable numbers](http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.0259v1.pdf) and [this discussion](http://tech.groups.yahoo.com/group/primenumbers/message/23021):



```
The asymptotic number of perfect numbers less than x was proven in

MR0090600 (19,837d)
Hornfeck, Bernhard; Wirsing, Eduard
Über die Häufigkeit vollkommener Zahlen. (German)
Math. Ann. 133 (1957), 431-438.

to be less than x^eps for all eps>0.
From this it trivially follows that the sum of the reciprocals
of the perfect numbers is finite.

```

