A136162 List of prime quadruplets {p, p+2, p+6, p+8}.

5, 7, 11, 13, 11, 13, 17, 19, 101, 103, 107, 109, 191, 193, 197, 199, 821, 823, 827, 829, 1481, 1483, 1487, 1489, 1871, 1873, 1877, 1879, 2081, 2083, 2087, 2089, 3251, 3253, 3257, 3259, 3461, 3463, 3467, 3469, 5651, 5653, 5657, 5659, 9431, 9433, 9437

Offset: 1

Author

Harry J. Smith, Dec 17 2007

Keywords

• nonn
• tabf

Comments

{11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q*(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - Jaroslav Krizek, Jul 07 2017

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Quadruplet

Crossrefs

Cf. A007530 (1st quadrisection).

Programs

Mathematica

Map[Prime[Range @@ #] &, MapAt[# + 1 &, SequencePosition[Differences@ Prime@ Range@ 1200, {2, 4, 2}], {All, -1}]] // Flatten (* Michael De Vlieger, Jul 11 2017 *)

PARI

{forprime(p1=0,70000,p2=p1+2;if(!isprime(p2),next;);p3=p1+6;if(!isprime(p3),next;);p4=p1+8;if(!isprime(p4),next;);print1(p1,",",p2,",",p3,",",p4,","))}

PARI

q=[0,0,0,0];i=0;forprime(p=5,1e4,(q[i%4+1]=p)==8+q[i++%4+1]&&print1(vecsort(q)","))  \\ M. F. Hasler, Apr 20 2013

Formula

[a(4n-3),a(4n-2),a(4n-1),a(4n)] = A007530(n) + [0,2,6,8], for all n>0. - M. F. Hasler, Apr 20 2013