  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.
