On Unique Additive Representations of Positive Integers and Some Close Problems

2008/11/03 by Shevelev, Vladimir
#11P81 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0811.0290

Abstract

Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. 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 (rm_r(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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