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

The 2-adic valuation of an integer n is the exponent of the highest power of 2 that divides n and is denoted by ν ₂(n) ν 2 ( n ) . In this paper, we prove that Euler’s partition function p ( n ) can be expressed in terms of ν ₂(n) ν 2 ( n ) . Our approach allows us to express the sum of positive divisors of n in terms of ν ₂(n) ν 2 ( n ) . 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.