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A128464 Numbers that are congruent to {11, 17, 29} mod 30. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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).

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

EXAMPLE

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

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 *)

PROG

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

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

CROSSREFS

Sequence in context: A038918 A220293 A166307 * A105170 A162175 A178128

Adjacent sequences:  A128461 A128462 A128463 * A128465 A128466 A128467

KEYWORD

easy,nonn

AUTHOR

Cino Hilliard, May 05 2007

EXTENSIONS

Better name by Omar E. Pol, Oct 28 2013

STATUS

approved

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Last modified May 25 04:36 EDT 2017. Contains 287008 sequences.