The sum of a digitaddition series

Abstract:

Let $B(x)$ be the number of ones in the binary expansion of $x$. A “digitaddition series” is a sequence ${y_1} < {y_2} < {y_3} < \ldots$, where ${y_1}$ is a given positive integer and ${y_{n + 1}} = {y_n} + B({y_n})\;{\text {for}}\;n = 1,2, \ldots$. Various questions involving the ${y_m}$ are studied; in particular, the asymptotic result ${y_m} \sim (m\log m)/(2\log 2)$ is proved.