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A052350 Least prime in A001359 (lesser of twin primes) such that the distance (A053319) to the next twin is 6*n.

%I A052350 #44 Mar 05 2025 01:53:35
%S A052350 5,17,41,617,71,311,2267,521,1877,461,1721,347,1151,1787,3581,2141,
%T A052350 6449,1319,21377,1487,12251,4799,881,23057,659,19541,12377,2381,38747,
%U A052350 10529,37361,8627,9041,33827,5879,80231,15359,45821,36107,14627,37991,36527,87251,70997
%N A052350 Least prime in A001359 (lesser of twin primes) such that the distance (A053319) to the next twin is 6*n.
%C A052350 Smallest distance (A052380) and also smallest possible increment of twin-distances is 6.
%C A052350 Primes may occur between p+2 and p+6n.
%C A052350 The prime a(n) determines a prime quadruple: [p, p+2, p+6n, p+6n+2] and a [2, 6n-2, 2] d-pattern.
%H A052350 Norman Luhn, <a href="/A052350/b052350.txt">Table of n, a(n) for n = 1..4624</a> (terms 1..4500 from Martin Raab).
%e A052350 The first 3 terms (5, 17, 41) are followed by difference patterns as it is displayed: 5 by [2, 4, 2], 17 by [2, 4+6, 2], 41 by [2, 4+6+6, 2] determining prime quadruples: (5, 7, 11, 13), (17, 19, 29, 31) or (41, 43, 59, 61), respectively.
%e A052350 a(10) = 461 gives the quadruple [461, 463, 521 = 461+60, 523], and between 521 and 463, 7 primes occur.
%t A052350 NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k += 6]; k];p = 5; t = Table[0, {50}]; Do[ q = NextLowerTwinPrim[p]; d = (q - p)/6; If[d < 51 && t[[d]] == 0, t[[d]] = p; Print[{d, p}]]; p = q, {n, 1500}]; t (* _Robert G. Wilson v_, Oct 28 2005 *)
%o A052350 (PARI) list(len) = {my(s = vector(len), c = 0, p1 = 5, q1 = 0, q2, d); forprime(p2 = 7, , if(p2 == p1 + 2, 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 A052350 Cf. A001359, A007530, A052380, A052381, A053319, A113274, A113275.
%Y A052350 Cf. A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.
%K A052350 nonn
%O A052350 1,1
%A A052350 _Labos Elemer_, Mar 07 2000
%E A052350 Name corrected by _Amiram Eldar_, Mar 04 2025