On Primitive 3-smooth Partitions of $n$

Abstract

A primitive 3-smooth partition of $n$ is a representation of $n$ as the sum of numbers of the form $2^a 3^b$, where no summand divides another. Partial results are obtained in the problem of determining the maximal and average order of the number of such representations. Results are also obtained regarding the size of the terms in such a representation, resolving questions of Erdős and Selfridge.