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A052352 Least prime in A031924 (lesser of 6-twins) such that the distance to the next 6-twin is 2*n.

%I A052352 #24 Mar 05 2025 01:54:52
%S A052352 47,23,73,61,353,31,233,131,331,653,2441,3733,1033,4871,1063,1621,503,
%T A052352 607,4211,7823,2287,83,383,1231,2903,5981,1123,173,11981,11833,1367,
%U A052352 2063,4723,19681,2207,2131,2713,9533,6571,1657,23081,15913,7013,14051,9967,22447
%N A052352 Least prime in A031924 (lesser of 6-twins) such that the distance to the next 6-twin is 2*n.
%C A052352 The increment of distance of 6-twins (A053321) is 2 (not 6), the smallest distance (A052380) is 6.
%C A052352 The middle gap 2n-6 may include primes, e.g., n = 12, a(12) = 653 and between 659 and 659 + 2*12 - 6 = 677, two primes occur (661 and 673).
%C A052352 a(n) = p yields a prime quadruple [p, p+6, p+2n, p+2n+6] with difference pattern [6, 2n-6, 6].
%H A052352 Amiram Eldar, <a href="/A052352/b052352.txt">Table of n, a(n) for n = 3..2000</a>
%e A052352 For n = 3, 4, 5,  the quadruples are [47, 53, 53, 59] (a triple), [23, 29, 31, 37], [73, 79, 83, 89] with 53 - 47 = 6, 31 - 23 = 8 and 83 - 73 = 10 twin distances.
%t A052352 seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 6] // Flatten; pp = p[[i]]; dd = Differences[pp]/2 - 2; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* _Amiram Eldar_, Mar 04 2025 *)
%o A052352 (PARI) list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 11, , if(p2 == p1 + 6, q2 = p1; if(q1 > 0, d = (q2 - q1)/2 - 2; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ _Amiram Eldar_, Mar 04 2025
%Y A052352 Cf. A031924, A052380, A052381, A053321.
%Y A052352 Cf. A052350, A052351, A052353, A052354, A052355, A052356, A052357, A052358, A052359.
%K A052352 nonn
%O A052352 3,1
%A A052352 _Labos Elemer_, Mar 07 2000
%E A052352 Name and offset corrected by _Amiram Eldar_, Mar 04 2025