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Eleven-free integers

Problem 442

Published on Saturday, 26th October 2013, 01:00 pm; Solved by 189

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).