Claims

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1975_01_01_erdos: Erdős's 1975 paper proves that every sum of reciprocals of positive multiples of 2^(2^k) is irrational, so 2^(2^n) is an irrationality sequence of the problem's kind.

1991_01_01_hancl: Hančl proves that an irrationality sequence must satisfy limsup of log_2 log_2 a_n over n at least 1, which with Erdős's example 2^(2^n) pins the slowest growth at doubly exponential.

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