0%

# Problem 616

## 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 $10^{12}$.

## 创意数

• 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$L$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}