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

3373, 32917, 2221, 13597, 3391, 37783, 4057, 13537, 8581, 41911, 6763, 7333, 10867, 12457, 1831, 2113, 14683, 37201, 6637, 17581, 25423, 37447, 11353, 11197, 20611, 22453, 57397, 1933, 50707, 37591, 11503, 39733, 2593, 122131, 22921, 9013, 17167, 10273, 9661

Offset: 3

Author

Labos Elemer, Mar 07 2000

Keywords

• nonn

Comments

The smallest distance between 16-twins [A052380(8)] is 18 and its minimal increment is 6.

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].

Examples

			a(9) = p = 4057 gives [4057, 4073, 4111, 4127] quadruple and [16, 38, 16] distance pattern with 4 primes in the medial gap.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 3..1002

Crossrefs

Cf. A031934, A052380, A052381, A053326.

Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052358, A052359.

Programs

Mathematica

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 *)

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

Extensions

Incorrect 43207 removed and more terms from Sean A. Irvine, Nov 06 2021

Name and offset corrected by Amiram Eldar, Mar 05 2025