Counting Flimsy Numbers via Formal Language Theory
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Let s_2(n) be the sum of the digits of n when expressed in base 2. For integers n and k, Stolarsky defined n to be k-flimsy if s_2(kn) < s_2(n). In this paper, we generalize the definition of k-flimsy numbers to all bases b, and provide a method to construct a pushdown automaton recognizing the k-flimsy base-b numbers. Using the tools of context-free languages and analytic combinatorics, we use this automaton to determine precise asymptotics for the number of k-flimsy N-digit numbers in base b. Lastly, using the results we obtained, we discuss the natural densities of k-flimsy numbers in base b for all values k and b. Our main results can be found in Theorems 2, 3, 8, and 9.
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Trevor Clokie (2021). Counting Flimsy Numbers via Formal Language Theory. UWSpace. http://hdl.handle.net/10012/16786