TITLE: A sequence that avoids both arithmetic and geometric progressions
QUESTION [58 upvotes]: Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link.
I started to explore sequences that avoid both arithmetic and geometric progressions,
i.e., avoid $x, x+c, x+2c$ and avoid $y, c y, c^2 y$ anywhere in the sequence
(not necessarily consecutively).
Starting with $(1,2)$, one cannot extend with $3$ because $(1,2,3)$ forms an
arithemtical progression, and one cannot extend with $4$ because $(1,2,4)$ is a geometric
progression. But $(1,2,5)$ is fine.
Continuing in the same manner leads to the following "greedy" sequence:
$$1, 2, 5, 6, 12, 13, 15, 16, 32, 33, 35, 39, 40, 42, 56, 81, 84, 85, 88,$$
$$90, 93, 94, 108, 109, 113, 115, 116, 159, 189, 207, 208, 222, \ldots$$
This sequence is not in the OEIS.
Here are a few questions:

Q1. What is its growth rate? 


 
 
 



Q2. Does $\sum_{i=1}^\infty 1/s_i$ converge? (Where $s_i$ is the $i$-th term of the above
  sequence.)
Q3. If it does, does it converge to e?
  Update: No. The sum appears to be approximately $2.73 > e$, as per
  @MichaelStocker and @Turambar.

That is wild numerical speculation. The first 457 terms (the extent
of the graph above) sum to 2.70261.

Addendum. 11Jul2014. Starting with $(0,1)$ rather than $(1,2)$ renders
a direct hit on OEIS A225571.

REPLY [2 votes]: Since both @awwalker and @mathworker21 mentioned Erdős' conjecture, and because
a paper discussing this conjecture was just published, I thought I would mention it:

Erdős Conjecture (1940s or 1950s). If $A \subset \mathbb{N}$ 
  satisfies $\sum_{n \in A} \frac{1}{n}= \infty$, then
  $A$ contains arbitrarily long arithmetic progressions.

In 

Grochow, Joshua. "New applications of the polynomial method: The cap
set conjecture and beyond." Bulletin of the American Mathematical
Society, Vol.56, No.1, Jan. 2019,

he says:

"It remains open even to prove that a set $A$ satisfying the hypothesis contains $3$-term arithmetic progressions."