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%I #22 Sep 08 2022 08:45:30
%S 11,17,29,41,47,59,71,77,89,101,107,119,131,137,149,161,167,179,191,
%T 197,209,221,227,239,251,257,269,281,287,299,311,317,329,341,347,359,
%U 371,377,389,401,407,419,431,437,449,461,467,479,491,497,509,521,527,539
%N Numbers that are congruent to {11, 17, 29} mod 30.
%C Numbers of the form 30k+r, 0 < r < 30, that are possible lower bounds of twin prime pairs.
%C 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.
%H Vincenzo Librandi, <a href="/A128464/b128464.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F From _Wesley Ivan Hurt_, Jun 14 2016: (Start)
%F G.f.: x*(11+6*x+12*x^2+x^3)/((x-1)^2*(1+x+x^2)).
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
%F a(n) = 10*n-1+4*sin(2*n*Pi/3)/sqrt(3).
%F a(3k) = 30k-1, a(3k-1) = 30k-13, a(3k-2) = 30k-19. (End)
%F 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
%e 41 = 30*1 + 11, the lower part of the twin prime pair 41,43.
%p 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
%t Select[Range[0, 800], MemberQ[{11, 17, 29}, Mod[#, 30]] &] (* _Wesley Ivan Hurt_, Jun 14 2016 *)
%t LinearRecurrence[{1, 0, 1, -1}, {11, 17, 29, 41}, 100] (* _Vincenzo Librandi_, Jun 15 2016 *)
%o (PARI) g(n) = forstep(x=11,n,30,print1(x","x+6","x+18","))
%o (Magma) [n : n in [0..800] | n mod 30 in [11, 17, 29]]; // _Wesley Ivan Hurt_, Jun 14 2016
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, May 05 2007
%E Better name by _Omar E. Pol_, Oct 28 2013
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