A Partial Asymptotic Formula for the Niven Numbers

A Niven number is a positive integer that is divisible by its digital sum. That is, if n is an integer and s(n) denotes the digital sum of n, then n is a Niven number if and only if sin) is a factor of n. This idea was introduced in [1] and investigated further in [2], [3], and [4]. One of the questions about the set N of Niven numbers was the status of -. . Njx) lim ——^5 x->-<» X where Nix) denotes the number of Niven numbers less than x. This limit, if it exists5 is called the "natural density" of N. It was proven in [3] that the natural density of the set of Niven numbers is zero, and in [4] a search for an asymptotic formula for N(x) was undertaken. That is, does there exist a function fix) such that