Euler’s partition function in terms of 2-adic valuation

Abstract

The 2-adic valuation of an integer n is the exponent of the highest power of 2 that divides n and is denoted by ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. In this paper, we prove that Euler’s partition function p(n) can be expressed in terms of ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. Our approach allows us to express the sum of positive divisors of n in terms of ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. We introduce the notion of 2-adic color partition and provide a new combinatorial interpretation for Euler partition function p(n). Connections between partitions and the game of m-Modular Nim with two heaps are presented in this context.