VWO 2000 Problem 3

Let $p_n$ be the $n$ -th prime. ( $p_1=2$ ) Define the sequence $(f_j)$ as follows:

• $f_1=1, f_2=2$
• $\forall j\ge 2$ : if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$
• $\forall j\ge 2$ : if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$

(a) Show that all $f_i$ are different (b) from which index onwards are all $f_i$ at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?