Any number with of a form $\frac{1}{n}$ has a decimal with a repetend of finite length that is never longer than $n$ (provable by Dirichlet principle). (Example: $\frac{92}{99}=0.929292\ldots$ in which case it is 92 that is repeating and the length of the series is 2.) Is there a way to find ALL numbers of the form $\frac{1}{n}$ which have repetends of length EXACTLY $n$? Are there infinitely many of them and does $\sum\_{\text{$n$ with this property}}\frac{1}{n}$ converge or not?



Is there a way to specify a special kind of reciprocals of natural numbers?
This is a textbook example of a question for which one should turn to the [OEIS](https://oeis.org/) for assistance. The first few elements of this set are


$$ 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, \dotsc$$


and the OEIS then tells you that these are the [full reptend primes](https://en.wikipedia.org/wiki/Full_reptend_prime) in base 10 ([OEIS A001913](https://oeis.org/A001913)). In fact just entering the first three elements 7, 17, 19 of this sequence into the OEIS will return the full reptend primes as the top search result.


[Artin's primitive root conjecture](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots) predicts that this set has asymptotic density
$$ \prod\_p \left(1 - \frac{1}{p(p-1)}\right) = 0.373955\dots$$
in the set of all primes; since the sum of reciprocals of primes diverges, it thus predicts that your sum also diverges. This conjecture is known ([Hooley - Artin's conjecture](https://mathscinet.ams.org/mathscinet-getitem?mr=207630)) under a sufficiently strong version of GRH, but remains open unconditionally; even the weaker statement that there are infinitely many such primes for a fixed base is unknown, though it is known for instance (see [Heath-Brown - Artin's conjecture for primitive roots](https://mathscinet.ams.org/mathscinet-getitem?mr=830627)) that there are at most two prime bases for which the latter claim fails.


