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

7, 67, 19, 43, 163, 127, 397, 229, 769, 1489, 673, 9547, 1009, 1783, 1693, 2857, 11677, 23869, 499, 1093, 4003, 28657, 10459, 29383, 12487, 6043, 41647, 7039, 17029, 19207, 15073, 24247, 65839, 29629, 18583, 9883, 66697, 100699, 7243, 53923, 82237, 6217, 76249

Offset: 1

Author

Labos Elemer, Mar 07 2000

Keywords

• nonn

Comments

a(n) is a "lesser of a 4-twin" prime whose distance to the next twin is 6n.

Both the smallest distance (A052380) and its increment for 4-twins is 6.

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].

Examples

			a(1) = 7 gives [7, 11,7+6 = 13, 17] with no primes between 11 and 13.
a(5) = 163 specifies [163, 167, 163+30 = 191, 193] with 4 primes between 167 and 193.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A023200, A052380, A052381, A053320.

Cf. A052350, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

Programs

Mathematica

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 *)

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

Extensions

Name corrected by Amiram Eldar, Mar 04 2025