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A052353 Least prime in A031926 (lesser of 8-twins) whose distance to the next 8-twin is 6*n.

%I A052353 #21 Mar 05 2025 01:55:23
%S A052353 389,683,719,359,1523,2699,401,929,2153,1373,2459,2531,1439,1733,8573,
%T A052353 2741,4943,9059,5051,983,3491,9173,7529,761,1823,1571,3041,5399,1193,
%U A052353 2273,491,8171,23549,5189,5813,53189,3221,4349,32789,49823,18749,19001,10979,89,19433
%N A052353 Least prime in A031926 (lesser of 8-twins) whose distance to the next 8-twin is 6*n.
%C A052353 The smallest distance [A052380(4)] between 8-twins is 12, while its minimal increment is 6.
%C A052353 a(n) = p yields a prime quadruple of [p, p+8, p+6n, p+6n+8] and difference pattern of [8, 6n-8, 8].
%H A052353 Amiram Eldar, <a href="/A052353/b052353.txt">Table of n, a(n) for n = 2..1001</a>
%e A052353 a(2) = 389 specifies quadruple of [389, 397, 401, 409] with no prime between 397 and 401;
%e A052353 a(11) = 1373 gives quadruple of [1373, 1381, 1439, 1447] and [8, 58, 8] difference pattern with 6 primes in the central gap.
%t A052353 seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 8] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 1; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* _Amiram Eldar_, Mar 05 2025 *)
%o A052353 (PARI) list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 8, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 1; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ _Amiram Eldar_, Mar 05 2025
%Y A052353 Cf. A031926, A052380, A052381, A053322.
%Y A052353 Cf. A052350, A052351, A052352, A052354, A052355, A052356, A052357, A052358, A052359.
%K A052353 nonn
%O A052353 2,1
%A A052353 _Labos Elemer_, Mar 07 2000
%E A052353 Name and offset corrected by _Amiram Eldar_, Mar 05 2025