This is a sketch of proof.
It is quite easy to prove this statement: "for n > 4, sigma(n) < prime(n)",
using an intermediate inequality like:
"for n > N, sigma(n) < C * n * ln n < prime(n)", where N is some fixed
integer and C a positive real constant as C(N).
We have to check the inequality when 4 < n <= N in order to complete
A note for the 1st inequality: it should be even stronger if the Riemann
Hypothesis holds - known as the Robin criterion: sigma(n) < C * n * ln (ln
but we need only: sigma(n) < C * n * ln n.
For the 2nd inequality: C * n * ln n < prime(n), use the behavior of the
inverse function of prime(n), the prime counting function pi(n) bounded
n / ln n < pi(n) < 1.25506 * n / ln n, if n >= 17. pi(n) ~ n / ln n.
We want: pi(floor(n * ln n)) < n, which is true because:
pi(floor(n * ln n)) < (n * ln n) / (ln (n * ln n)) < n.