Let $\Phi_n(x)$ be the $n$ - th cyclotomic polynomial.
Are there Carmichael numbers of the form $\Phi_n(b)$ , where $b$ is a positive integer and $n$ is a positive integer different from the numbers $1,2,3,4,6,10$ ?
The cases $\ n=1\ $ and $\ n=2\ $ trivially produce all Carmichael-numbers (in fact every positive integer expect $\ 1\ $ in the case $\ n=2\ $) , the cases $\ n=3\ $ and $\ n=6\ $ are equivalent because of $\ \Phi_3(x)=\Phi_6(x+1)\ $ . A survey of Carmichael numbers I have found so far is
gp > for(m=2,10^8,s=polcyclo(3,m);if(Mod(2,s)^(s-1)==1,if(ispseudoprime(s)==0,if(carm(s)==1,print(m," ",s," ",component(factor(s),1)~)))))
2304 5310721 [13, 37, 61, 181]
47735 2278677961 [7, 19, 103, 181, 919]
97944 9593125081 [331, 2311, 12541]
172799 29859667201 [7, 13, 31, 61, 73, 2377]
gp > for(m=2,10^8,s=polcyclo(4,m);if(Mod(2,s)^(s- 1)==1,if(ispseudoprime(s)==0,if(carm(s)==1,print(m," ",s," ",component(factor(s),1)~)))))
216 46657 [13, 37, 97]
1560 2433601 [17, 37, 53, 73]
8208 67371265 [5, 13, 37, 109, 257]
592956 351596817937 [17, 37, 197, 433, 6553]
649800 422240040001 [13, 37, 41, 73, 241, 1217]
4321800 18677955240001 [29, 61, 101, 197, 701, 757]
21415680 458631349862401 [37, 53, 89, 19009, 138241]
gp >
The only other Carmichael number I found is
10 27 512461 [31, 61, 271]
In particular, I found no Carmichael number for $\ n=5\ $ and for $\ n>6\ $ except $\ n=10\ $
Are there other Carmichael numbers of this form , in particular for the missing $\ n\ $ and is there another Carmichael number for $\ n=10\ $ ? Are there useful necessary conditions to speed up the search ?