Divisibility and distribution of mex-related integer partitions of Andrews and Newman

Article Type

Research Article

Publication Title

International Journal of Number Theory

Abstract

Andrews and Newman introduced the minimal excludant or "mex"function for an integer partition π of a positive integer n, mex(π), as the smallest positive integer that is not a part of π. They defined σmex(n) to be the sum of mex(π) taken over all partitions π of n. We prove infinite families of congruence and multiplicative formulas for σmex(n). By restricting to the part of π , Andrews and Newman also introduced moex(π) to be the smallest odd integer that is not a part of π and σmoex(n) to be the sum of moex(π) taken over all partitions π of n. In this paper, we show that for any sufficiently large X, the number of all positive integer n ≤ X such that σmoex(n) is an even (or odd) number is at least (loglog X).

First Page

581

Last Page

592

DOI

https://10.1142/S1793042123500288

Publication Date

4-1-2023

Comments

Open Access, Green

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