Algebraic Entropies of Natural Numbers With One or Two Prime Factors

Abstract

Abstract. We formulate the additive entropy of a natural number in terms of the additive partition function, and show that its multiplicative entropy is directly related to the multiplicative partition function. We give a practical formula for the multiplicative entropy of natural numbers with two prime factors. We use this formula to analyze the comparative density of additive and multiplicative entropy, prove that this density converges to zero as the number tends to infinity, and empirically observe this asymptotic behavior.

About the Entropy of a Natural Number and a Type of the Entropy of an Ideal

“…As a result, we have a discrete probability distribution associated with a natural number. Similarly, there are some studies related to the entropy of a natural number—namely, Jeong et al, in [ 3 ], defined the additive entropy of a natural number in terms of the additive partition function, and in [ 4 ], we found the following definition for the entropy of a natural number: where is the sum of natural divisors of n . Additionally, regarding the entropy H of a natural number, introduced in [ 5 ], another type of entropy is a natural number.…”

“…We have found several ways to define the entropy of a natural number. Jeong et al, in [ 3 ], defined the additive entropy of a natural number in terms of the additive partition function. If d is the divisor of a natural number n , then we will write .…”