A128464 Numbers that are congruent to {11, 17, 29} mod 30.

11, 17, 29, 41, 47, 59, 71, 77, 89, 101, 107, 119, 131, 137, 149, 161, 167, 179, 191, 197, 209, 221, 227, 239, 251, 257, 269, 281, 287, 299, 311, 317, 329, 341, 347, 359, 371, 377, 389, 401, 407, 419, 431, 437, 449, 461, 467, 479, 491, 497, 509, 521, 527, 539

Offset: 1

Author

Cino Hilliard, May 05 2007

Keywords

• easy
• nonn

Comments

Numbers of the form 30k+r, 0 < r < 30, that are possible lower bounds of twin prime pairs.

For a 30k+r "wheel", r = 11,17,29 are the only possible values that can form a lower twin prime pair. The 30k+r wheel gives the recurrence 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, ... which is frequently used in prime number sieves to skip multiples of 2, 3, 5. The fact that adding 2 to 30k+1, 7, 13, 19, 23 will gives us a multiple of 3 or 5, precludes these numbers from being a lower member of a twin prime pair. This leaves us with r = 11, 17, 29 as the only possible cases to form a lower bound of a twin prime pair.

Examples

			41 = 30*1 + 11, the lower part of the twin prime pair 41,43.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Programs

Magma

[n : n in [0..800] | n mod 30 in [11, 17, 29]]; // Wesley Ivan Hurt, Jun 14 2016

Maple

A128464:=n->10*n-1+4*sin(2*n*Pi/3)/sqrt(3): seq(A128464(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Mathematica

Select[Range[0, 800], MemberQ[{11, 17, 29}, Mod[#, 30]] &] (* Wesley Ivan Hurt, Jun 14 2016 *)
LinearRecurrence[{1, 0, 1, -1}, {11, 17, 29, 41}, 100] (* Vincenzo Librandi, Jun 15 2016 *)

PARI

g(n) = forstep(x=11,n,30,print1(x","x+6","x+18","))

Formula

From Wesley Ivan Hurt, Jun 14 2016: (Start)

G.f.: x*(11+6*x+12*x^2+x^3)/((x-1)^2*(1+x+x^2)).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = 10*n-1+4*sin(2*n*Pi/3)/sqrt(3).

a(3k) = 30k-1, a(3k-1) = 30k-13, a(3k-2) = 30k-19. (End)

E.g.f.: 1 + (10*x - 1)*exp(x) + 4*sin(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2))/sqrt(3). - Ilya Gutkovskiy, Jun 15 2016

Extensions

Better name by Omar E. Pol, Oct 28 2013