The irrationality of a divisor function series of Erdős and Kac

For positive integers $k$ and $n$ let ${\sigma }_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erdős and Kac asked whether, for every $k$, the number ${\alpha }_k={\sum }_{n\ge 1}\frac{{\sigma }_k(n)}{n!}$ is irrational. It is known unconditionally that ${\alpha }_k$ is irrational if $k\le 3$. We prove ${\alpha }_4$ is irrational.