4,1

The smallest distance between 20-twins is 24 [= A052380(10)], while its minimal increment is 6.

a(n) = p starts [p, p+20, p+6n, p+6n+20] and [20, 6n-20, 20] patterns of primes and their difference.

a(n) = p is the smallest prime which starts a [p, p+20] twin followed by the next [p+6n, p+6n+20] twin.

Amiram Eldar, Table of n, a(n) for n = 4..1003

For n = 4, a(4) = 46703 results in prime quadruple [46703, 46723, 46727, 46747] and difference pattern [20, 4, 20].

For n = 14, a(14) = 5903 yields prime quadruple [5903, 5923, 5987, 6007] with 4 primes in the medial gap, and difference pattern [20, 64, 20].

seq[m_] := Module[{p = Prime[Range[m]], d, i, pp, dd, j}, d = Differences[p]; i = Position[d, 20] // Flatten; pp = p[[i]]; dd = Differences[pp]/6 - 3; j = TakeWhile[FirstPosition[dd, #] & /@ Range[Max[dd]] // Flatten, ! MissingQ[#] &]; pp[[j]]]; seq[15000] (* 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 + 20, q2 = p1; if(q1 > 0, d = (q2 - q1)/6 - 3; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2); } \\ Amiram Eldar, Mar 05 2025

Cf. A031938, A052380, A052381, A053331.

Cf. A052350, A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358.

Sequence in context: A265285 A255514 A251502 * A205752 A205474 A232411

Adjacent sequences: A052356 A052357 A052358 * A052360 A052361 A052362

Labos Elemer, Mar 07 2000

Offset changed to 1 by Michel Marcus, Apr 30 2019

Name and offset corrected by Amiram Eldar, Mar 05 2025

approved