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Multi-base representations of integers: Asymptotic enumeration and central limit theorems

Krenn Daniel (Alpen-Adria-Universität Klagenfurt, Department of Mathematics, Klagenfurt, Austria)
Ralaivaosaona Dimbinaina (Stellenbosch University, Department of Mathematical Sciences, Matieland, South Africa)
Wagner Stephan (Stellenbosch University, Department of Mathematical Sciences, Matieland, South Africa)

In a multi-base representation, in contrast to the common b-ary representation, the base is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, meaning that each integer can have multiple different digit expansions. We provide a general asymptotic formula for the number of multi-base representations of a positive integer. Moreover, we prove central limit theorems for different statistics associated to a multi-base representation.

Keywords: multi-base representations, asymptotic formula, partitions, central limit theorem