8 Minimal Niven numbers

Define a k to be the smallest positive multiple of k such that the sum of its digits in base q is equal to k. The asymptotic behavior, lower and upper bound estimates of a k are investigated. A characterization of the minimality condition is also considered. A positive integer n is a Niven number (or a Harshad number) if it is divisible by the sum of its (decimal) digits. For instance, 2007 is a Niven number since 9 divides 2007. A q-Niven number is an integer k which is divisible by the sum of its base q digits, call it s q (k) (if q = 2, we shall use s(k) for s 2 (k)). Niven numbers have been extensively studied by various just to cite a few of the most recent works). In this paper, we define a natural sequence in relation to q-Niven numbers. For a fixed but arbitrary k ∈ N and a base q ≥ 2, one may ask whether or not there exists a q-Niven