The existence of n-flimsy numbers in a given base

Abstract

Let the function \(s_g\) map a positive integer to the sum of its digits in the base g. A number k is called n-flimsy in the base g if \(s_g(nk)<s_g(k)\). Clearly, given a base g, \(g\geqslant 2\), if n is a power of g, then there does not exist an n-flimsy number in the base g. We give a constructive proof of the existence of an n-flimsy number in the base g for all the other values of n (such an existence follows from the results of Schmidt and Steiner, but the explicit construction is a novelty). Our algorithm for construction of such a number, say k, is very flexible in the sense that, by easy modifications, we can impose further requirements on k—k ends with a given sequence of digits, k begins with a given sequence of digits, k is divisible by a given number (or belongs to a certain congruence class modulo a given number), etc.