829. Integral Fusion

Given any integer $n>1$$n \gt 1$ a binary factor tree $T(n)$$T(n)$ is defined to be:

• A tree with the single node $n$$n$ when $n$$n$ is prime.

• A binary tree that has root node $n$$n$, left subtree $T(a)$$T(a)$ and right subtree $T(b)$$T(b)$, when $n$$n$ is not prime. Here $a$$a$ and $b$$b$ are positive integers such that $n=ab$$n = ab$, $a\le b$$a\le b$ and $b-a$$b-a$ is the smallest.

For example $T(20)$$T(20)$:

We define $M(n)$$M(n)$ to be the smallest number that has a factor tree identical in shape to the factor tree for $n!!$$n!!$, the double factorial of $n$$n$.

For example, consider $9!!=9\times 7\times 5\times 3\times 1=945$$9!! = 9\times 7\times 5\times 3\times 1 = 945$. The factor tree for $945$$945$ is shown below together with the factor tree for $72$$72$ which is the smallest number that has a factor tree of the same shape. Hence $M(9)=72$$M(9) = 72$.

Find ${\displaystyle \sum _{n=2}^{31}M(n)}$$\displaystyle\sum_{n=2}^{31} M(n)$.

829. 整数合成

对于任意整数 $n>1$$n \gt 1$，其二叉因子树 $T(n)$$T(n)$ 定义如下：

• $n$$n$ 是质数时，该树只有 1 个节点 $n$$n$。

• $n$$n$ 不是质数时，$T(n)$$T(n)$ 的根为 $n$$n$，左子树、右子树分别为 $T(a)$$T(a)$、$T(b)$$T(b)$。其中正整数 $a,b$$a, b$ 满足 $n=ab$$n=ab$，$a\le b$$a \leq b$，且 $b-a$$b - a$ 最小。

如下图为 $T(20)$$T(20)$：

记 $M(n)$$M(n)$ 为满足如下条件的最小整数：该数的二叉因子树与 $n!!$$n!!$（$n$$n$ 的双阶乘）的二叉因子树形状完全一致。如下图，$n=9$$n = 9$ 时，$9!!=9\times 7\times 5\times 3\times 1=945$$9!! = 9\times 7\times 5\times 3\times 1 = 945$。$945$$945$ 的二叉因子树与 $72$$72$ 的二叉因子树形状完全一致，而且 $72$$72$ 是满足此条件的最小正整数，故 $M(9)=72$$M(9) = 72$。

求 ${\displaystyle \sum _{n=2}^{31}M(n)}$$\displaystyle\sum_{n=2}^{31} M(n)$。

点 这个链接 回到源站。

点 这个链接 回到详细版题目目录。