A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.

Entries in chains are congruent to {1,7,3,9} mod 10.

Examples

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.
		

Links

• John Cerkan, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A085956, A086361, A086362, A023330, A059766.

Subsequence of A007693, A023256, and A024899.

Programs

Magma

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010

Mathematica

k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

Formula

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Extensions

Additional comments from Labos Elemer, Jul 23 2003