cp's OEIS Frontend

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

%I A182518 #12 Feb 16 2025 08:33:13
%S A182518 63973,31146661,703995733,21595159873,192739365541,461574735553,
%T A182518 3976486324993,10028704049893,84154807001953,197531244744661,
%U A182518 741700610203861,973694665856161,2001111155103061,3060522900274753,3183276534603733,4271903575869601
%N A182518 Carmichael numbers of the form C = p*(2p-1)*(3p-2)*(6p-5), where p is prime.
%C A182518 We get Carmichael numbers with four prime divisors for p = 7, 271, 337, 727, 1237, 1531, 2281, 3037, 3067.
%C A182518 We get Carmichael numbers with more than four prime divisors for p = 31, 67, 157, 577, 2131, 2731, 3301.
%C A182518 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).
%H A182518 Charles R Greathouse IV, <a href="/A182518/b182518.txt">Table of n, a(n) for n = 1..10000</a>
%H A182518 E. W. Weisstein, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%o A182518 (PARI) search(lim)={
%o A182518     my(v=List(),n,f);
%o A182518     forprime(p=7,lim,
%o A182518         n=p*(2*p-1)*(3*p-2)*(6*p-5)-1;
%o A182518         if(n%(p-1),next);
%o A182518         f=factor(2*p-1);
%o A182518         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182518         f=factor(3*p-2);
%o A182518         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182518         f=factor(6*p-5);
%o A182518         for(i=1,#f[,1],if(f[i,2]>1 || n%(f[i,1]-1), next(2)));
%o A182518         listput(v,n+1)
%o A182518     );
%o A182518     Vec(v)
%o A182518 }; \\ _Charles R Greathouse IV_, Oct 02 2012
%K A182518 nonn
%O A182518 1,1
%A A182518 _Marius Coman_, May 03 2012