A182518 Carmichael numbers of the form C = p*(2p-1)*(3p-2)*(6p-5), where p is prime.

63973, 31146661, 703995733, 21595159873, 192739365541, 461574735553, 3976486324993, 10028704049893, 84154807001953, 197531244744661, 741700610203861, 973694665856161, 2001111155103061, 3060522900274753, 3183276534603733, 4271903575869601

Offset: 1

Author

Marius Coman, May 03 2012

Keywords

• nonn

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

Programs

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