On the distribution of self-numbers

Abstract:

Self-numbers are those integers which cannot be expressed as $a + f(a)$, where $f(a)$ denotes the sum of the digits of $a$ in a given scale. Here I prove that the number of self-numbers less than or equal to a large number $x$ equals $Lx + O({\log ^2}x)$, where $L$ is a strictly positive constant.