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.
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The research was supported by the Ministry of Science and Technological Development of Serbia (Project 174006) and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina (Project “Ordered structures and applications”).
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Bašić, B. The existence of n-flimsy numbers in a given base. Ramanujan J 43, 359–369 (2017). https://doi.org/10.1007/s11139-015-9768-7
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DOI: https://doi.org/10.1007/s11139-015-9768-7