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A101186 Values of k for which 7m+1, 8m+1 and 11m+1 are prime, with m = 1848k + 942.

%I A101186 #19 Apr 27 2024 03:35:57
%S A101186 13,123,218,223,278,411,513,551,588,733,743,796,856,928,1168,1226,
%T A101186 1263,1401,1533,1976,1981,2013,2096,2138,2241,2376,2556,2676,2703,
%U A101186 3626,3703,3718,3971,4008,4121,4138,4163,4188,4211,4313,4423,4653,4656,4901,5018
%N A101186 Values of k for which 7m+1, 8m+1 and 11m+1 are prime, with m = 1848k + 942.
%C A101186 The number (7m+1)(8m+1)(11m+1) is a 3-factor Carmichael number if and only if m is equal to 1848k+942 with k in this sequence. The sequence includes the value k = 10^329 - 4624879 which yields a 1000-digit Carmichael number with three prime factors of 334 digits each. Other Carmichael numbers of the same form would necessarily have 4 prime factors or more; the smallest such example is 3664585=127*(7*29)*199, for m=18.
%H A101186 Robert Israel, <a href="/A101186/b101186.txt">Table of n, a(n) for n = 1..10000</a>
%H A101186 Gérard P. Michon, <a href="http://www.numericana.com/answer/modular.htm#chernik">Generic Carmichael Numbers</a>.
%e A101186 a(1)=13 because k=13 corresponds to m=24966, which yields a product of three primes (7m+1)(8m+1)(11m+1) equal to the Carmichael number 9585921133193329. (Among all Carmichael numbers with 16 or fewer digits, as first listed by Richard G. E. Pinch, this one features the largest "least prime factor".)
%p A101186 filter:= proc(n) local m;
%p A101186   m:= 1848*n+942;
%p A101186   andmap(isprime,[7*m+1,8*m+1,11*m+1])
%p A101186 end proc:
%p A101186 select(filter, [$1..10000]); # _Robert Israel_, May 14 2019
%t A101186 q[k_] := Module[{m = 1848*k + 942}, PrimeQ[7*m + 1] && PrimeQ[8*m + 1] && PrimeQ[11*m + 1]]; Select[Range[6000], q] (* _Amiram Eldar_, Apr 27 2024 *)
%o A101186 (Magma) [k:k in [1..5100]| forall{s:s in [7,8,11]|IsPrime(m*s+1) where m is 1848*k+942}]; // _Marius A. Burtea_, Nov 01 2019
%o A101186 (PARI) is(k) = {my(m = 1848*k + 942); isprime(7*m + 1) && isprime(8*m + 1) && isprime(11*m + 1);} \\ _Amiram Eldar_, Apr 27 2024
%Y A101186 Cf. A002997 (Carmichael numbers), A046025.
%K A101186 nonn
%O A101186 1,1
%A A101186 _Gerard P. Michon_, Dec 03 2004