A052354 Least prime in A031928 (lesser of 10-twins) whose distance to the next 10-twin is 6*n.

409, 691, 787, 547, 2053, 139, 4861, 283, 181, 25087, 337, 709, 2917, 829, 14197, 919, 3001, 33589, 2767, 421, 8221, 1879, 5179, 1249, 1471, 10141, 5011, 20533, 4483, 54091, 13249, 4663, 27883, 5869, 41443, 8599, 23311, 9049, 40699, 82591, 3109, 5323, 44917, 11971

Offset: 2

Author

Labos Elemer, Mar 07 2000

Keywords

• nonn

Comments

a(n) = p determines a prime quadruple [p, p+10, p+6n, p+6n+10] with difference pattern [10, 6n-10, 10].

The smallest distance between 10-twins [A052380(5)] is 12, while its increment is 6.

a(n) = p is the smallest of A031928 followed by the next 10-twin after a 6n distance.

Examples

			a(3) = 691 results in [691, 701, 709, 719] quadruple and [10, 8, 10] difference pattern without primes in the median gap.
a(11) = 25087 yields [25087, 25097, 25153, 25163] and [10, 56, 10] with 5 primes in the middle gap.
		

Links

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

Crossrefs

Cf. A031928, A052380, A052381, A053323.

Cf. A052350, A052351, A052352, A052353, 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, 10] // 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 + 10, 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