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A052351 Least prime in A023200 (lesser of 4-twins) such that the distance to the next 4-twin is 6*n.

%I A052351 #22 Mar 05 2025 01:54:21
%S A052351 7,67,19,43,163,127,397,229,769,1489,673,9547,1009,1783,1693,2857,
%T A052351 11677,23869,499,1093,4003,28657,10459,29383,12487,6043,41647,7039,
%U A052351 17029,19207,15073,24247,65839,29629,18583,9883,66697,100699,7243,53923,82237,6217,76249
%N A052351 Least prime in A023200 (lesser of 4-twins) such that the distance to the next 4-twin is 6*n.
%C A052351 a(n) is a "lesser of a 4-twin" prime whose distance to the next twin is 6n.
%C A052351 Both the smallest distance (A052380) and its increment for 4-twins is 6.
%C A052351 The prime a(n)=p is the first which determines a prime quadruple [p, p+4, p+6n, p+6n+4] and difference pattern of [4, 6n-4, 4].
%H A052351 Amiram Eldar, <a href="/A052351/b052351.txt">Table of n, a(n) for n = 1..1000</a>
%e A052351 a(1) = 7 gives [7, 11,7+6 = 13, 17] with no primes between 11 and 13.
%e A052351 a(5) = 163 specifies [163, 167, 163+30 = 191, 193] with 4 primes between 167 and 193.
%t A052351 seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 4] // Flatten; pp = p[[i]]; dd = Differences[pp]/6; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* _Amiram Eldar_, Mar 04 2025 *)
%o A052351 (PARI) list(len) = {my(s = vector(len), c = 0, p1 = 7, q1 = 0, q2, d); forprime(p2 = 11, , if(p2 == p1 + 4, q2 = p1; if(q1 > 0, d = (q2 - q1)/6; 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 A052351 Cf. A023200, A052380, A052381, A053320.
%Y A052351 Cf. A052350, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.
%K A052351 nonn
%O A052351 1,1
%A A052351 _Labos Elemer_, Mar 07 2000
%E A052351 Name corrected by _Amiram Eldar_, Mar 04 2025