TITLE: Carmichael number divisible by a given positive integer
QUESTION [11 upvotes]: Prove or disprove this conjecture : If $k>2$ is an integer with $\gcd(k,\varphi(k))=1$ , then there is a Carmichael number $N$ with $k\mid N$

The condition $\gcd(k,\varphi(k))=1$ is necessary. Otherwise, $k$ would not be squarefree and therefore also $N$ , which is impossible. Or there would be prime factors $p,q$ of $k$ and therefore $N$, with $p\mid q-1$. This is also impossible since this would imply $p\mid N-1$ because of $q-1\mid N-1$ contradicting $p\mid N$
But is it also sufficient ? And if yes, is there an efficient method to construct a Carmichael number divisible by $k$ ? A particular hard case seems to be $k=885=3\cdot 5\cdot 59$.

Is there a Carmichael number divisble by $885$ ?

REPLY [2 votes]: The smallest Carmichael number that is a multiple of 885, is:
3399464645479365585

Below 2^64, there are only two more Carmichael numbers that are divisible by 885:
15165761619468172065
15885147377097813585

It's an open-problem whether every odd cyclic number has at least one Carmichael multiple.
See also:

https://oeis.org/A253595
https://www.numericana.com/data/crump.htm

Several methods for constructing Carmichael numbers (including source-code implementations), are listed at:

https://trizenx.blogspot.com/2020/08/pseudoprimes-construction-methods-and.html