An irrationality sequence is a sequence of positive integers 
such that 
converges
to an irrational number for every sequence 
of positive
integers. Erdős (1975, pp. 6-7) showed that the sequence 
, whose terms for 
, 1, 2, ... are 2, 4, 16, 256, 65536, ... (OEIS A001146),
is an irrationality sequence (Guy 1994, p. 225).
Irrationality Sequence
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References
Erdős, P. "Some Problems and Results on the Irrationality of the Sum of Infinite Series." J. Math. Sci. 10, 1-7, 1975. https://www.renyi.hu/~p_erdos/1976-44.pdf.Guy, R. K. "Irrationality Sequences." §E24 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 225, 1994.Sloane, N. J. A. Sequence A001146/M1297 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Irrationality SequenceCite this as:
Weisstein, Eric W. "Irrationality Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IrrationalitySequence.html