Prime Numbers

1.5 Exercises 57
(4) For odd n, investigate the possibility of “close calls” to perfection. For
example, show (by machine perhaps) that every odd n with 10 1, the set of integers n with k|σ(n)
has asymptotic density 1. (Hint: Use the Dirichlet Theorem 1.1.5.) The
case k = 4 is easier than the general case. Use this easier case to show
that the set of odd perfect numbers has asymptotic density 0.
(7) Let s(n) =σ(n) − n for natural numbers n, andlets(0) = 0. Thus, n is
abundant if and only if s(n) >n.Lets (k) (n) be the function s iterated k
times at n. Use the Dirichlet Theorem 1.1.5 to prove the following theorem
of H. Lenstra: For each natural number k thereisanumbern with
n