Is it known whether a constant $A$ exists, such that there is always a Carmichael-number between $x$ and $Ax$ for $x\ge 561$?

Question

For prime numbers, such a constant is well known. For $n\ge 2$, there is always a prime between $n$ and $2n$, so $A=2$ does the job.

Carmichael-numbers are much rarer, so I wonder, whether a constant $A$ has been found and be proven to do the job for $n\ge 561$ ?

The count upto $10^{21}$ shows that there are Carmichael-numbers with $3,4,5,...,21$ digits, so is $A=10$ sufficient ?

Answer 1

Carmichael-numbers are not so rare as it perhaps seems. If $C(x)$ denotes the number of Carmichael numbers $\le x$, then Harman proved in $2005$ that $C(x)>x^{0.332}$ for $x$ sufficiently large. Erdös has even conjectured that there are $x^{1-o(1)}$ Carmichael numbers for $x$ sufficiently large, although present computations do not really point to this. From the asymptotic density one could obtain such a constant $A$.