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

%I A136162 #25 Feb 16 2025 08:33:07
%S A136162 5,7,11,13,11,13,17,19,101,103,107,109,191,193,197,199,821,823,827,
%T A136162 829,1481,1483,1487,1489,1871,1873,1877,1879,2081,2083,2087,2089,3251,
%U A136162 3253,3257,3259,3461,3463,3467,3469,5651,5653,5657,5659,9431,9433,9437
%N A136162 List of prime quadruplets {p, p+2, p+6, p+8}.
%C A136162 {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
%H A136162 Michael De Vlieger, <a href="/A136162/b136162.txt">Table of n, a(n) for n = 1..10000</a>
%H A136162 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadruplet.html">Prime Quadruplet</a>
%F A136162 [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
%t A136162 Map[Prime[Range @@ #] &, MapAt[# + 1 &, SequencePosition[Differences@ Prime@ Range@ 1200, {2, 4, 2}], {All, -1}]] // Flatten (* _Michael De Vlieger_, Jul 11 2017 *)
%o A136162 (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,","))}
%o A136162 (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
%Y A136162 Cf. A007530 (1st quadrisection).
%K A136162 nonn,tabf
%O A136162 1,1
%A A136162 _Harry J. Smith_, Dec 17 2007