A052353 Least prime in A031926 (lesser of 8-twins) whose distance to the next 8-twin is 6*n.

389, 683, 719, 359, 1523, 2699, 401, 929, 2153, 1373, 2459, 2531, 1439, 1733, 8573, 2741, 4943, 9059, 5051, 983, 3491, 9173, 7529, 761, 1823, 1571, 3041, 5399, 1193, 2273, 491, 8171, 23549, 5189, 5813, 53189, 3221, 4349, 32789, 49823, 18749, 19001, 10979, 89, 19433

Offset: 2

Author

Labos Elemer, Mar 07 2000

Keywords

• nonn

Comments

The smallest distance [A052380(4)] between 8-twins is 12, while its minimal increment is 6.

a(n) = p yields a prime quadruple of [p, p+8, p+6n, p+6n+8] and difference pattern of [8, 6n-8, 8].

Examples

			a(2) = 389 specifies quadruple of [389, 397, 401, 409] with no prime between 397 and 401;
a(11) = 1373 gives quadruple of [1373, 1381, 1439, 1447] and [8, 58, 8] difference pattern with 6 primes in the central gap.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..1001

Crossrefs

Cf. A031926, A052380, A052381, A053322.

Cf. A052350, A052351, A052352, 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, 8] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 1; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[10000] (* Amiram Eldar, Mar 05 2025 *)

PARI

list(len) = {my(s = vector(len), c = 0, p1 = 2, q1 = 0, q2, d); forprime(p2 = 3, , if(p2 == p1 + 8, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 1; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 05 2025

Extensions

Name and offset corrected by Amiram Eldar, Mar 05 2025