A specific sequence such that there is no three-term arithmetic progression in the sequence: does the corresponding series of reciprocals diverge? Define the sequence of natural numbers $(a_n)_n$ recursively as follows:
For each $k\geq 0,\ $ define $a_{k+1}$ to be the least natural number such that it doesn't make a three term arithmetic progression with (two) previous terms in the sequence.
So if I haven't made a mistake, this sequence is: $1,2,4,5,10,11,13,14,28,29, 31, 32, 37, 38, 40, 41,\ldots.$
My question is, does $\sum \frac{1}{a_k}$ converge or diverge?
I believe this question is related to Erdos' conjecture on arithmetic progressions, which I have been reading about.

If I'm not mistaken, $a_n$ grows roughly as $n^\alpha$ where $\alpha = \log_2(3) \approx 1.585$.  Since $\alpha > 1$, $\sum_n 1/a_n$ converges.
