A340573 a(n) is the smallest lesser twin prime p from A001359 such that the distance to the previous lesser twin prime is 6*n.

11, 29, 59, 641, 101, 347, 2309, 569, 1931, 521, 1787, 419, 1229, 1871, 3671, 2237, 6551, 1427, 21491, 1607, 12377, 4931, 1019, 23201, 809, 19697, 12539, 2549, 38921, 10709, 37547, 8819, 9239, 34031, 6089, 80447, 15581, 46049, 36341, 14867, 38237, 36779, 87509, 71261, 15137, 40427, 13679, 54917, 41141, 50891

Offset: 1

Author

Artur Jasinski, Jan 12 2021

Keywords

• nonn

Comments

Lesser twin primes (with the exception of prime 3) are congruent to 5 modulo 6, which implies that distances between successive pairs of twin primes are 6*k.

Examples

			a(1)=11 because 11 - 5 = 6*1.
a(2)=41 because 41 - 29 = 6*2.
a(3)=59 because 59 - 41 = 6*3.
		

Links

• Martin Raab, Table of n, a(n) for n = 1..4508

Crossrefs

Cf. A001359, A001359, A053319, A007530, A052380, A052381, A113274, A113275, A052350.

Programs

Mathematica

Table[a[n] = 0, {n, 1, 10000}]; Table[
b[n] = 0, {n, 1, 10000}]; qq = {}; prev = 5; Do[
If[Prime[n + 1] - Prime[n] == 2, k = (Prime[n] - prev)/6;
  If[b[k] == 0, a[k] = Prime[n]; b[k] = 1]; prev = Prime[n]], {n, 5,
  10000}]; list = Table[a[n], {n, 1, 50}]
(* Second program: *)
pp = Select[Prime[Range[10^4]], PrimeQ[#+2]&];
dd = Differences[pp];
a[n_] := pp[[FirstPosition[dd, 6n][[1]]+1]];
Array[a, 50] (* Jean-François Alcover, Jan 13 2021 *)

Formula

a(n) = A052350(n) + 6*n.