MathDB

3

Part of 2000 Flanders Math Olympiad

Problems(1)

prime sequence

Source: flanders '00

9/27/2005
Let pnp_n be the nn-th prime. (p1=2p_1=2) Define the sequence (fj)(f_j) as follows: - f1=1,f2=2f_1=1, f_2=2 - ∀j≥2\forall j\ge 2: if fj=kpnf_j = kp_n for k<pnk<p_n then fj+1=(k+1)pnf_{j+1}=(k+1)p_n - ∀j≥2\forall j\ge 2: if fj=pn2f_j = p_n^2 then fj+1=pn+1f_{j+1}=p_{n+1} (a) Show that all fif_i are different (b) from which index onwards are all fif_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?
number theoryprime numbers