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Problem 442

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Eleven-free integers

An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of 11 except 1.

For example, 2404 and 13431 are eleven-free, while 911 and 4121331 are not.

Let E(n) be the nth positive eleven-free integer. For example, E(3) = 3, E(200) = 213 and E(500 000) = 531563.

Find E(10¹⁸).

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无11整数

无11整数指的是十进制表达式中不包含有除了1之外11的幂的整数。

例如，2404和13431都是无11整数，而911和4121331不是。

记E(n)是第n个无11正整数。例如，E(3) = 3，E(200) = 213，以及E(500 000) = 531563。

求E(10¹⁸)。

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