OFFSET
1,1
COMMENTS
We get Carmichael numbers with four prime divisors for p = 7, 271, 337, 727, 1237, 1531, 2281, 3037, 3067.
We get Carmichael numbers with more than four prime divisors for p = 31, 67, 157, 577, 2131, 2731, 3301.
Note: we can see that p, 2p-1, 3p-2 and 6p-5 can all four be primes only for p = 6k+1 (for p = 6k+5, we get 2p-1 divisible by 3), so in that case the formula is equivalent to C = (6k+1)(12k+1)(18k+1)(36k+1).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. W. Weisstein, Carmichael Number
PROG
(PARI) search(lim)={
my(v=List(), n, f);
forprime(p=7, lim,
n=p*(2*p-1)*(3*p-2)*(6*p-5)-1;
if(n%(p-1), next);
f=factor(2*p-1);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(3*p-2);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
f=factor(6*p-5);
for(i=1, #f[, 1], if(f[i, 2]>1 || n%(f[i, 1]-1), next(2)));
listput(v, n+1)
);
Vec(v)
}; \\ Charles R Greathouse IV, Oct 02 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, May 03 2012
STATUS
approved