标题： 喜悦数 显示英文标题

标题： Elated Numbers

摘要： 对于基数$b \geq 2$，$b$相关函数$E_{2,b}$将以基数$b$表示的正整数映射到其首位数字与各位数字平方和的乘积。 一个$b$相关数是在迭代$E_{2,b}$下映射到$1$的正整数。 一个$b$-elated 数的高是将其映射到$1$所需的迭代次数。我们确定了$E_{2,b}$的不动点和循环，并证明了关于$b$-elated 数和$b$-elated 数的最小高度序列的一系列结果。尽管$b$-elated 函数与$b$-happy 函数密切相关，但两者的行为明显不同，这由本工作中的结果所证明。 显示英文摘要

摘要： For a base $b \geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the number of iterations required to map it to $1$. We determine the fixed points and cycles of $E_{2,b}$ and prove a range of results concerning sequences of $b$-elated numbers and $b$-elated numbers of minimal heights. Although the $b$-elated function is closely related to the $b$-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work.