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
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, May 05 2007
EXTENSIONS
Better name by Omar E. Pol, Oct 28 2013
STATUS
approved