1952/10/01 by A. Sklar · 20 citations
Mathematics · Computer Science · Engineering · #Advanced Mathematical Theories #Coding theory and cryptography #graph theory and CDMA systems #Square-free integer #Factorization #Mathematics #Combinatorics #Algorithm
paper · pdf · doi:10.1090/s0002-9939-1952-0050620-1
published in Proceedings of the American Mathematical Society 3(5), 701-705 (American Mathematical Society)
openalex publication_date 1952/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/03/11
In recent years several papers [1; 3; 4; 5; 6; 7; 9; 10; 11] have appeared dealing with the problem of Factorisatio numerorum, the number f(n) of representations of an integer n as an ordered product of factors greater than 1. As a result, the basic combinatorial properties of f(n) and the asymptotic behavior of its summatory function are well known. In this paper, I determine the asymptotic behavior of f(n) itself for squarefree n and use the result to determine a normal order of f(n) for all n. If n is written as a product of powers of distinct primes pi, then f(n) is evidently a symmetric function of the exponents of the pi's. If n is squarefree, all the exponents are 1 and f(n) can be considered a function of the single variable r, the number of distinct prime factors of n. It is therefore convenient to define, for positive integral r: