On the counting function for the Niven numbers

Counting the number of twin Niven numbers

“…As in [4], given k ∈ N, t ∈ N 0 and an integer , we set A(x|k, , t) := #{0 ≤ n < x : n ≡ (mod k) and α(n) = t}. Now consider the positive integers t satisfying log x t log x and, for such t's, define…”

“…Moreover, because t ≡ j (mod q(q − 1)), we have that ( t , q − 1) = ( j , q − 1) and [ t , q] = t ·q ( j ,q) . Consequently, using (4) and Lemma 3, it follows from (6) that…”

On the counting function for the generalized Niven numbers

“…which was proven only recently by De Koninck, Doyon and Kátai [6], and independently by Mauduit, Pomerance and Sárközy [8].…”

“…Most of the proof of Theorem 1 is a straightforward generalization of the methods used in [6] and it is the intent of this paper to indicate where significant and perhaps non-obvious modifications to that work must be made. The first notable change is the introduction and use of the quantities d and F .…”

“…Given an arbitrary non-zero, integer-valued, completely q additive function f , we define an f -Niven number to be a positive integer n that is divisible by f (n). In the final section of [6] it is suggested that the techniques used therein could be applied to derive an asymptotic expression for the counting function of the f -Niven numbers,…”