Note on translated sum on primitive sequences

In this note, we construct a new set \boldsymbol{S} of primitive sets such that for any real number x\geq 60 we get: \begin{equation*} \sum\limits_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}>\sum\limits_{p\in \mathcal{P}}\frac{1}{p(\log p+x)},\text{ }\mathcal{A\in }{\boldsymbol{S}}, \end{equation*} where \mathcal{P} denotes the set of prime numbers.

The degree of primitive sequences and Erdős conjecture

A sequence A of strictly positive integers is said to be primitive if none of its term divides another. Z. Zhang proved a result, conjectured by Erdős and Zhang in 1993, on the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 4. In this paper, we extend this result to the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 5.

On a translated sum over primitive sequences related to a conjecture of Erdős

For x large enough, there exists a primitive sequence \mathcal{A}, such that \begin{equation*} \sum\limits_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}\gg \sum\limits_{p\in \mathcal{P}}\frac{1}{p(\log p+x)}\text{,} \end{equation*} where \mathcal{P} denotes the set of prime numbers.

Extracting Initial Iterative Control Signal Based on Trajectory Primitives Matching and Combining

The initial iterative control signal is often adopted a zero or a certain value in the conventional iterative learning control (ILC) system, and an ILC process needs to renew again as long as the desired trajectory is changed. In this paper, the NURBS (Non-Uniform Rational B-Splines) model is used for describing all trajectory primitives and the desired trajectory. It is studied that the problem of the initial iterative control signal is extracted in ILC, which presents a method of extracting the initial iterative control signal based on the trajectory primitive optimal matching and combining algorithm. Firstly, the definition of the similarity index between the two different spacial trajectories is introduced. Secondly, an optimal matching and combining algorithm is designed under a certain similarity index, which is used to find two or more combined primitive sequences with space patterns similar to the desired trajectory. Thirdly, the initial iterative control signals of the desired trajectory are extracted by using the control information of the combined primitive sequences. Finally, the simulation is carried out to demonstrate the effectiveness of the present method.