论文标题
在$ k $的数字上
On $k$-layered numbers
论文作者
论文摘要
如果可以将其除数分配为$ k $套件，则据说一个正整数$ n $被划分为$ k $。在本文中，我们开始对这些数字类别的系统研究。特别是，我们指出一些算法以找到一些甚至$ k $的数字$ n $，因此对于每个正整数$α$，$ 2^αn$都是$ k $的数字。我们还找到了最小的$ k $ aylered号码，价格为$ 1 \ leq k \ leq 8 $。此外，我们在$ n！$是$ 3 $的时候学习，何时是$ 4 $ aylered的数字。此外，我们将所有$ 4 $覆盖的数字分类为$ n = p^p^αq^βrt$，其中$α$，$ 1 \ leqβ\ leq 3 $，$ p $，$ q $，$ r $和$ t $分别为两个正整数和四个普莱姆。此外，在本文中，讨论了有关这些数字的其他一些结果及其与$ k $ - 多种型数字，接近完美的数字和超量数字的关系。此外，我们发现每个正整数$ 1 \ leq k \ leq 5 $的连续$ k $ aylered数字的差异。最后，通过假设最小的$ k $ aylayer数字，我们找到了一个连续两个$ k $ aylaered号码差的上限。
A positive integer $n$ is said to be $k$-layered if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we start the systematic study of these class of numbers. In particular, we state some algorithms to find some even $k$-layered numbers $n$ such that $2^αn$ is a $k$-layered number for every positive integer $α$. We also find the smallest $k$-layered number for $1\leq k\leq 8$. Furthermore, we study when $n!$ is a $3$-layered and when is a $4$-layered number. Moreover, we classify all $4$-layered numbers of the form $n=p^αq^βrt$, where $α$, $1\leq β\leq 3$, $p$, $q$, $r$, and $t$ are two positive integers and four primes, respectively. In addition, in this paper, some other results concerning these numbers and their relationship with $k$-multiperfect numbers, near-perfect numbers, and superabundant numbers are discussed. Also, we find an upper bound for the differences of two consecutive $k$-layered numbers for every positive integer $1\leq k\leq 5$. Finally, by assuming the smallest $k$-layered number, we find an upper bound for the difference of two consecutive $k$-layered numbers.
