Abstract
AbstractThe 2-adic valuation of an integer n is the exponent of the highest power of 2 that divides n and is denoted by $$\nu _2(n)$$
ν
2
(
n
)
. In this paper, we prove that Euler’s partition function p(n) can be expressed in terms of $$\nu _2(n)$$
ν
2
(
n
)
. Our approach allows us to express the sum of positive divisors of n in terms of $$\nu _2(n)$$
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.
Publisher
Springer Science and Business Media LLC
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