Irrationality sequence

The powers of two whose exponents are powers of two, 2^{2^n}, form an irrationality sequence. However, although Sylvester's sequence :2, 3, 7, 43, 1807, 3263443, ... (in which each term is one more than the product of all previous terms) also grows doubly exponentially, it does not form an irrationality sequence. For, letting x_n=1 for all n gives :\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots = 1, a series converging to a rational number. Likewise, the factorials, n!, do not form an irrationality sequence because the sequence given by x_n=n+2 for all n leads to a series with a rational sum, :\sum_{n=0}^{\infty}\frac{1}{(n+2)n!}=\frac{1}{2}+\frac{1}{3}+\frac{1}{8}+\frac{1}{30}+\frac{1}{144}+\cdots=1. ==Growth rate==

For any sequence an to be an irrationality sequence, it must grow at a rate such that :\limsup_{n\to\infty} \frac{\log\log a_n}{n} \geq \log 2 . This includes sequences that grow at a more than doubly exponential rate as well as some doubly exponential sequences that grow more quickly than the powers of powers of two. ==Related properties==

Analogously to irrationality sequences, has defined a transcendental sequence to be an integer sequence an such that, for every sequence xn of positive integers, the sum of the series : \sum_{n=1}^\infty \frac{1}{a_n x_n} exists and is a transcendental number. ==References==