Asymptotic distribution of integers with certain prime factorizations

Localized Factorizations of Integers

We determine the order of magnitude of H(k+1)(x,y, 2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1, . . . , log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear i...

Asymptotic Distribution of Divergence Measure with Applications

On the Asymptotic Distribution of Certain Sums

Delange’s Tauberian theorem and asymptotic normality of random ordered factorizations of integers

By a suitable shifting-the-mean parametrization at the Dirichlet series level and Delange’s Tauberian theorems, we show that the number of factors in random ordered factorizations of integers is asymptotically normally distributed.

Ordered and Unordered Factorizations of Integers

We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having all factors distinct. We implement all these m...