cp's OEIS Frontend

A052357 Least prime in A031934 (lesser of 16-twins) whose distance to the next 16-twin is 6*n.

%I A052357 #30 Mar 05 2025 01:58:34
%S A052357 3373,32917,2221,13597,3391,37783,4057,13537,8581,41911,6763,7333,
%T A052357 10867,12457,1831,2113,14683,37201,6637,17581,25423,37447,11353,11197,
%U A052357 20611,22453,57397,1933,50707,37591,11503,39733,2593,122131,22921,9013,17167,10273,9661
%N A052357 Least prime in A031934 (lesser of 16-twins) whose distance to the next 16-twin is 6*n.
%C A052357 The smallest distance between 16-twins [A052380(8)] is 18 and its minimal increment is 6.
%C A052357 a(n) = p is the smallest prime introducing the prime quadruple [p, p+16, p+6n, p+6n+16], which has a difference pattern [16, 6n-16, 16].
%H A052357 Amiram Eldar, <a href="/A052357/b052357.txt">Table of n, a(n) for n = 3..1002</a>
%e A052357 a(9) = p = 4057 gives [4057, 4073, 4111, 4127] quadruple and [16, 38, 16] distance pattern with 4 primes in the medial gap.
%t A052357 seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 16] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 2; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[12000] (* _Amiram Eldar_, Mar 05 2025 *)
%o A052357 (PARI) list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 16, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 2; 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 A052357 Cf. A031934, A052380, A052381, A053326.
%Y A052357 Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052358, A052359.
%K A052357 nonn
%O A052357 3,1
%A A052357 _Labos Elemer_, Mar 07 2000
%E A052357 Incorrect 43207 removed and more terms from _Sean A. Irvine_, Nov 06 2021
%E A052357 Name and offset corrected by _Amiram Eldar_, Mar 05 2025