This is somewhat related to Erdős conjecture on arithmetic progressions

Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ contains arbitrary long arithmetic progressions?

If one allows negative integers a solution is $a_n=(-1)^n n$

`$\{a_n\}$`

and a very fastly growing integer sequence`$\{b_n\}$`

. In succession, add $b_1$ to all the terms of $a_n$ after the first, add $b_2$ to all the terms after the third, ..., add $b_r$ to all the terms after the $\binom{r+1}{2}$-th. If $b_n$ is at least (roughly) $n^4$, say $b_n=((n!)!)!$, this should work. – M P Jan 19 '12 at 10:56