The distribution of square-free numbers

On Square-Free Numbers

In the article the formal characterization of square-free numbers is shown; in this manner the paper is the continuation of [19]. Essentially, we prepared some lemmas for convenient work with numbers (including the proof that the sequence of prime reciprocals diverges [1]) according to [18] which were absent in the Mizar Mathematical Library. Some of them were expressed in terms of clusters’ re...

Counting Square-Free Numbers

The main topic of this contribution is the problem of counting square-free numbers not exceeding n. Before this work we were able to do it in time Õ( √ n). Here, the algorithm with time complexity Õ(n) and with memory complexity Õ(n) is presented. Additionally, a parallel version is shown, which achieves full scalability. As of now the highest computed value was for n = 10. Using our implementa...

On the Average Sensitivity of Testing Square-Free Numbers

determination of maximal singularity free zones in the workspace of parallel manipulator

due to the limiting workspace of parallel manipulator and regarding to finding the trajectory planning of singularity free at workspace is difficult, so finding a best solution that can develop a technique to determine the singularity-free zones in the workspace of parallel manipulators is highly important. in this thesis a simple and new technique are presented to determine the maximal singula...

On the Square Roots of Triangular Numbers

1. BALANCING NUMBERS We call an Integer n e Z a balancing number if 1+ 2+ --+ (»l ) = (w + l) + (w + 2) +••• + (» + >•) (1) for some r e Z. Here r is called the balancer corresponding to the balancing number n. For example, 6, 35, and 204 are balancing numbers with balancers 2, 14, and 84, respectively. It follows from (1) that, if n is a balancing number with balancer r, then n2^(n + r)(n + r ...