Does the upper bound hold for every positive integer $n$?

Question

Mathworld gives an upper bound for the number of Carmichael numbers below $n $ here

Does this upper bound hold for all positive integers $n$ ?

What is meant with "sufficiently large" concerning the lower bound ? Do we know a reasonable small $n$ doing the job ?

Answer 1

For the lower bound:

In "Prime numbers a computational perspective" they say

The "sufficiently large" in Theorem 3.4.7 [stating $C(x) > x^{2/7}$] has not been calculated, but probably it is the 96th Carmichael number, $8719309$. From calculations in [Pinch 1993] it seems likely that $C(x)>x^{1/3}$ for all $x\geq 10^{15}$. $\dots$ Though Erdős has conjectued that $C(x) > x^{1-\varepsilon}$ for $x \geq x_0(\varepsilon)$, we know no numerical value of $x$ with $C(x)>x^{1/2}$.

Note that the book is a little dated and not everything is up to date, so the values may have changed.

And for the upper bound they also say "sufficiently" large values of $x$. They do, however, reference Pomerance 1981. From what I gather, they only worried about asymptotics, not actual values.