3 prime factor Charmichael number examples for roughly every 3 decimal digits up to 343

Car_3.343dd.gp

assert(b)=if(!(b),error());
factmul(f1,f2)=matreduce(matconcat([f1,f2]~));
factval(F)=vecprod([v[1]^v[2]|v<-F~]);

{Redu=[0,25,0,25,110,291,51,146,131,511,111,95,1121,2685,820,12481,16175,1866,
4500,11525,8960,441,390,14796,1280,1651,1730,24140,21226,18555,43391,3716,2980,
46701,38580,15450,5560,19445,14376,83660,32560,7516,5060,23806,57806,44636,
28985,73445,60936,55146,91400,82190,54255,8016,25591,71945,259946,147035,11301,
3375,2371,18486,466191,436551,422806,6220,153406,493275,222755,1572896,453141,
5385,422511,663666,364225,84081,52590,916505,285466,827301,5671,137266,120160,
23755,202690,140935,503151,560981,1528285,2474481,527805,660190,391711,
1161251,2985,727031,984236,494371,311091,517321,289351,496466,143956,33130,
1394135,166086,484375,4787525,2336340,338141,50106,411670,822656,507900];}
Car_3_m=Redu;for(i=1,#Redu,Car_3_m[i]=10^(i-1)+Redu[i]);

Car_3_pqr=[pqr|m<-Car_3_m;p<-[6*m+1];q<-[12*m+1];r<-[18*m+1];pqr<-[[p,q,r]]];
Car_3_n=[n|pqr<-Car_3_pqr;n<-[pqr[1]*pqr[2]*pqr[3]]];
Car_3_npqr=[[n,p,q,r]|i<-[1..#Car_3_n];n<-[Car_3_n[i]];\
                      pqr<-[Car_3_pqr[i]];p<-[pqr[1]];q<-[pqr[2]];r<-[pqr[3]]];

Quick=Set(concat([1..38],\
          [41,42,43,45,47,48,50,51,54,55,59,67,70,73,76,79,81,93,111,114]));
Carfactorint(i)={
  if(i==80,
    return([2,4;3,5;29,1;101,1;2879,1;3527,1;794593,1;154660403,1;1375778713,1;
      82559127467,1;2191141520677,1;5244008545913,1;215523228663966371,1;
      727945109737073604951121,1;3071482538062826439493217,1;
      405473649637074755617787593281337701367,1;
      357632907601311950933336045572973824205399821454828478880978313,1]
    )
  );
  [n,p,q,r]=Car_3_npqr[i];
  np=(n-1)/gcd(n-1,p-1);npq=np/gcd(np,q-1);npqr=npq/gcd(npq,r-1);
  F=factmul(factorint((n-1)/npqr),factorint(npqr));
  assert(n-1==factval(F));
  F
};

All 3-Carmichael numbers of this gist are product of 3 primes of the form 6*m+1, 12*m+1 and 18*m+1.
They are stored compressed in vector Redu.
The uncompressed m-values are in vector Car_3_m.
The corresponding prime tuples are in vector Car_3_pqr.
The corresponding Carmichael numbers are in vector Car_3_n.
For convenience vector Car_3_npqr gives access to [n,p,q,r] tuples.

These are the number of decimal digits of the available Carmichael numbers , up to 343:

hermann@7950x:~$ gp -q Car_3.343dd.gp 
? version
[2, 17, 3]
? foreach(Car_3_n,n,print1(#digits(n)","))
4,8,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,133,136,139,142,145,148,151,154,157,160,163,166,169,172,175,178,181,184,187,190,193,196,199,202,205,208,211,214,217,220,223,226,229,232,235,238,241,244,247,250,253,256,259,262,265,268,271,274,277,280,283,286,289,292,295,298,301,304,307,310,313,316,319,322,325,328,331,334,337,340,343,
? 

This verifies that all p,q,r are primes, and that n is product of the corresponding p,q,r:

? #
   timer = 1 (on)
? foreach(Car_3_npqr,npqr,[n,p,q,r]=npqr;print1(n==p*q*r&&isprime(p)&&isprime(q)&&isprime(r)))
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
cpu time = 6,896 ms, real time = 1,731 ms.
? 

Basic feature of Carmichael numbers n is, that n is composite and despite that for each base a coprime to n expression 1==lift(Mod(a,n)^(n-1)) is true (n is a Fermat pseudoprime to base a). Here demonstrated for top 10 Carmichael numbers from this gist, and 20 different bases:

? m=#Car_3_n;for(i=m-9,m,n=Car_3_n[i];for(a=n-21,n-2,print1(lift(Mod(a,n)^(n-1))==1));print(" ",#digits(n)))
11111111111111111111 316
11111111111111111111 319
11111111111111111111 322
11111111111111111111 325
11111111111111111111 328
11111111111111111111 331
11111111111111111111 334
11111111111111111111 337
11111111111111111111 340
11111111111111111111 343
cpu time = 87 ms, real time = 87 ms.
? 

I created this gist to have Carmichael numbers of reasonable size, that allow for fast factorization of n-1.
Not all of this gist Carmichael numbers allow for that.
Indices allowing fast (less than 3s) factorization (with npqr trick in Carfactorint()) are in vector Quick.
There are holes, but even for big numbers every 40 decimal digits a next fast to factor can be found:

? foreach(Quick,i,print1(#digits(Car_3_n[i])","))
4,8,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,64,67,70,73,76,79,82,85,88,91,94,97,100,103,106,109,112,115,124,127,130,136,142,145,151,154,163,166,178,202,211,220,229,238,244,280,334,343,
? 

Confirmation that factorization times are indeed less than 3000ms for all indices in Quick:

? foreach(Quick,i,gettime();Carfactorint(i);print1(gettime()","))
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,38,0,2,1,89,117,685,1,13,16,429,231,2364,1470,0,130,3,3,413,610,14,1795,316,0,9,80,52,327,565,1179,2028,523,426,0,1,2156,1408,
cpu time = 17,490 ms, real time = 17,491 ms.
? 

Nested infinite loops used to generate the compressed vector Redu entries.
Terminated after 2.5h real and 8.5h cpu time (PARI/GP isprime()is multithreaded):

good(m)=isprime(6*m+1)&&isprime(12*m+1)&&isprime(18*m+1);
for(d=0,oo,for(m=10^d,oo,if(good(m),print1(m-10^d,",");break))); 

Efforts to determine other factorizations of n-1 for Carmichael numbers n:
https://www.mersenneforum.org/node/1106355?p=1106709#post1106709

New index 80 fully factorized, 63 decimal digits prime is largest prime factor of 241 decimal digits n-1:

hermann@j4105:~$ gp -q Car_3.343dd.gp
? n=Car_3_n[80];
? #digits(n)
241
? F=Carfactorint(80);
? ##
  ***   last result computed in 0 ms.
? n-1==factval(F)
1
? for(i=1,#F-1,if(F[i,1]>=F[i+1,1],print("wrong "i)))
? [isprime(p)|p<-F[,1]]
time = 1,088 ms.
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
? print(Carfactorint(80))
[2, 4; 3, 5; 29, 1; 101, 1; 2879, 1; 3527, 1; 794593, 1; 154660403, 1; 1375778713, 1; 82559127467, 1; 2191141520677, 1; 5244008545913, 1; 215523228663966371, 1; 727945109737073604951121, 1; 3071482538062826439493217, 1; 405473649637074755617787593281337701367, 1; 357632907601311950933336045572973824205399821454828478880978313, 1]
?

Chernick, Jack. “On Fermat's simple theorem.” Bulletin of the American Mathematical Society 45 (1939): 269-274.
https://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf

There are more forms of Carmichael numbers with 3 primes: