# Sylvester's sequence

Sylvester's sequence $a_1,a_2,a_3,\ldots$ is defined recursively by setting $a_1=2$ and $a_k = a_1 \ldots a_{k-1}+1$ for all subsequent k, thus the sequence begins
There is a connection to the finding primes project: It is a result of Odoni that the number of primes less than n that can divide any one of the $a_k$ is $O(n / \log n \log\log\log n)$ rather than $O(n / \log n)$ (the prime number theorem bound). If we then factor the first k elements of this sequence, we must get a prime of size at least $k\log k \log \log \log k$ or so.
It is also conjectured that this sequence is square-free; if so, $a_k, a_k-1$ form a pair of square-free integers, settling a toy problem in the finding primes project.