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Problem 616

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Creative numbers

Alice plays the following game, she starts with a list of integers $L$ and on each step she can either:

• remove two elements $a$ and $b$ from $L$ and add $a^b$ to $L$
• or conversely remove an element $c$ from $L$ that can be written as $a^b$, with $a$ and $b$ being two integers such that $a,b>1$, and add both $a$ and $b$ to $L$

For example starting from the list $L=8$, Alice can remove $8$ and add $2$ and $3$ resulting in $L=2,3$ in a first step. Then she can obtain $L=9$ in a second step.

Note that the same integer is allowed to appear multiple times in the list.

An integer $n>1$ is said to be creative if for any integer $m>1$ Alice can obtain a list that contains $m$ starting from $L=n$.

Find the sum of all creative integers less than or equal to ${10}^{12}$.

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创意数

爱丽丝正在这样一个游戏：她从整数列表$L$开始，每一步她可以：

• 从$L$中去除两个元素$a$和$b$，并将$a^b$加入$L$
• 或者相反地，从$L$中去除一个可以写成$a^b$形式的元素$c$，其中$a$和$b$均为整数且$a,b>1$，然后将$a$和$b$加入$L$

例如，从整数列表$L=8$开始，第一步爱丽丝可以去除$8$并加入$2$和$3$，得到$L=2,3$。然后第二步她可以得到$L=9$。

注意在列表中同一整数允许重复出现多次。

称整数$n>1$为创意数，如果对于任意整数$m>1$，爱丽丝都能从$L=n$开始得到一个包含有$m$的整数列表。

找出所有小于或等于${10}^{12}$的创意数之和。

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