pernumber

Pernumber is a mathematical construct introduced in analytic number theory to quantify the distributional characteristics of integers with respect to their prime composition. The term, which is a portmanteau of “per” and “number,” refers to a function that assigns to each positive integer a value reflecting the density of prime factors in a normalized manner. In its most common formulation, the pernumber function \(P(n)\) is defined as the sum of the reciprocals of the logarithms of all distinct prime divisors of \(n\), that is,

\(P(n)=\sum_{p\mid n}\frac{1}{\log p}\).

If a prime factor occurs with multiplicity, it is counted only once. This normalization by the logarithm ensures that primes of larger magnitude contribute less, yielding a more balanced measure across scales.

The pernumber function shares similarities with the divisor sum function and the number‐of–prime‐factors function \(\omega(n)\), but its logarithmic weighting allows it to be more amenable to analytic techniques. It is additive over coprime factors, meaning \(P(ab)=P(a)+P(b)\) whenever \(\gcd(a,b)=1\). Consequently, \(P(n)\) can be expressed as a Dirichlet convolution of the characteristic function of the primes with the reciprocal logarithm function.

Applications of the pernumber concept occur in the study of the distribution of smooth numbers—integers whose prime factors are below a certain bound—as well as in sieve theory, where it aids in estimating the density of numbers with restricted prime factorization patterns. In cryptographic contexts, pernumber-related sums arise when analyzing the statistical properties of RSA moduli and in the evaluation of collision probabilities for hash functions based on integer factorization.

Variants of the pernumber have been explored, including weighted versions that replace the reciprocal logarithm with other smooth functions to emphasize different ranges of prime factors. The investigation of pernumber growth rates and the distribution of its values across the integers remains an active area of research in theoretical mathematics.