BG-MO 1973 Problem 1

Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$ . Prove that:

(a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$ . (b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$ I. Tonov