Irrationality sequence

In mathematics, an irrationality sequence is a sequence of positive integers a_n with the property that for any sequence x_n of positive integers, the sum of the series

$\sum _{n=1}^{\infty }{\frac {1}{a_{n}x_{n}}}$

exists and is an irrational number.

Examples of irrationality sequences include $2^{2^{n}}$. Any sequence a_n with

$\lim \sup {\frac {\log \log a_{n}}{n}}>\log 2$

is an irrationality sequence.

The sequence n! is not an irrationality sequence.

References

• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. E24. ISBN 0-387-20860-7. Zbl 1058.11001.