< n . Thus, f (f (n)) ≤ n for any n ≥ 7 and therefore any
sequence starting with n ≥ 7 is bounded by (n +1) and hence is periodic.Solution :
If a1 = 1, 2, 3, 4, 5, or 6 then the sequence is obviously periodic. Since for all natural numbers a and b exceeding 2, ab ≥ a + b , we have
f(n) ≤ n + 1 for all natural values of n.
Let n > 7 be even. Then f(n) = 2 + p2 + … + pr
+ 1 ≤ 3 + p2 • p3 • … • pr ≤ 3 +
< n . Thus, f (f (n)) ≤ n for any n ≥ 7 and therefore any
sequence starting with n ≥ 7 is bounded by (n +1) and hence is periodic.