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
};

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: