# nt.number theory – Is the factorization of \$a_m-a_n\$ affected by the fact that \$Sigma frac{1}{a_k}

I would like to ask the following.

Let $$(a_n)$$ be a sequence of natural numbers such that
$$sum_{k=1}^{infty}frac{1}{a_k}$$ converges. Is it true that for
infinitely many $$m$$, there is a $$n such that $$a_m-a_n$$ has a prime
divisor greater than $$m$$?

In other words, is it true that if for every $$m, n$$, the difference $$a_m-a_n$$ has all it’s prime factors less than or equal to $$m$$, then $$sum_{k=1}^{infty}frac{1}{a_k}=+infty$$?