Computational aspects of sturdy and flimsy numbers

• Published in: Theoretical computer science Vol. 927; pp. 65 - 86
• Main Authors: Clokie, Trevor, Lidbetter, Thomas F., Molina Lovett, Antonio, Shallit, Jeffrey, Witzman, Leon
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 26-08-2022
• Subjects: Context-free grammar; Enumeration; Finite automaton; Flimsy number; Sturdy number; Finite automaton; Enumeration; Sturdy number; Context-free grammar; Flimsy number
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2022.05.029

Following Stolarsky, we say that a natural number n is flimsy in base b if some positive multiple of n has smaller digit sum in base b than n does; otherwise it is sturdy. When n is proven flimsy by multiplier k, we say n is k-flimsy. We study computational aspects of sturdy and flimsy numbers.
We provide some criteria for determining whether a number is sturdy. We study the computational problem of checking whether a given number is sturdy, giving several algorithms for the problem, focusing particularly on the case b=2.
We find two additional, previously unknown sturdy primes. We develop a method for determining which numbers with a fixed number of 0's in binary are flimsy. Finally, we develop a method that allows us to estimate the number of k-flimsy numbers with n bits, and we provide explicit results for k=3 and k=5. Our results demonstrate the utility (and fun) of creating algorithms for number theory problems, based on methods of automata theory.
•We give a number of algorithms for determining if a given number is flimsy or sturdy.•We give a new method, based on automata theory, for counting the number of sturdy numbers.•We find two new, previously unknown sturdy prime numbers.