Abstract
Abstract The aim of this note is to study the distribution function of certain sequences of positive integers, including, for example, Bell numbers, factorials and primorials. In fact, we establish some general asymptotic formulas in this regard. We also prove some limits that connect these sequences with the number e. Furthermore, we present a generalization of the primorial.
Highlights
In combinatorics, the Bell number Bn counts the number of different ways to partition a set with n elements, where n ∈ N ∪ {0}
Bell numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell (1883–1960), Scottish mathematician, who wrote some comprehensive papers about them in the 1930s
We establish some new generalizations and we show that the relation (1.1) is true for some other known sequences as the sequence of factorials and the sequence of primorials
Summary
The Bell number Bn counts the number of different ways to partition a set with n elements, where n ∈ N ∪ {0}. Jakimczuk more generally showed that if a strictly increasing sequence of integers such as Fn satisfy the asymptotic formula log Fn ∼ cn log n (c > 0), the number of Fn that do not exceed n is asymptotically equivalent to log n c log log n as n We shall prove a general theorem on the distribution function of certain sequences of fast increase. If φ(x) is the distribution function of the sequence An (i.e., φ(x) is the number of An not exceeding x), the following asymptotic formula holds: (2.2)
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