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A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character sums.

In this paper, the consensus of heterogeneous multi-agent systems (MASs) with uncertain Deny-of-Service (DoS) attack strategies is studied. In our system, all agents are time synchronized and they communicate with each other with a constant sampling period normally. When the system is under attack, all agents use the hold-input mechanism to update the control protocol. By assuming that the attack duration is upper bounded and the occurrence of the attack follows a Markovian jumping process, the...