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

13, 123, 218, 223, 278, 411, 513, 551, 588, 733, 743, 796, 856, 928, 1168, 1226, 1263, 1401, 1533, 1976, 1981, 2013, 2096, 2138, 2241, 2376, 2556, 2676, 2703, 3626, 3703, 3718, 3971, 4008, 4121, 4138, 4163, 4188, 4211, 4313, 4423, 4653, 4656, 4901, 5018

Offset: 1

Author

Gerard P. Michon, Dec 03 2004

Keywords

• nonn

Comments

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.

Examples

			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".)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Gérard P. Michon, Generic Carmichael Numbers.

Crossrefs

Cf. A002997 (Carmichael numbers), A046025.

Programs

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

Maple

filter:= proc(n) local m;
  m:= 1848*n+942;
  andmap(isprime,[7*m+1,8*m+1,11*m+1])
end proc:
select(filter, [$1..10000]); # Robert Israel, May 14 2019

Mathematica

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 *)

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