On Unique Additive Representations of Positive Integers and Some Close Problems
Abstract
Let, for r>=2, (mr(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form sk+rsl. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion relation for Moser's numbers. We also study interesting properties of the sequence (rmr(n-1)+1),n>=1, and its connection with some important problems. In particular, in the case of r=2 this sequence is surprisingly connected with the numbers solving the combinatorial Josephus-Groer problem. We pose also some open questions.
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