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A230462 Numbers congruent to {1, 11, 13, 17, 19, 29} mod 30. 2

%I

%S 1,11,13,17,19,29,31,41,43,47,49,59,61,71,73,77,79,89,91,101,103,107,

%T 109,119,121,131,133,137,139,149,151,161,163,167,169,179,181,191,193,

%U 197,199,209,211,221,223,227,229,239,241,251,253,257,259,269

%N Numbers congruent to {1, 11, 13, 17, 19, 29} mod 30.

%C Reduces sieving for all twin primes (A001097) except (3,5) and (5,7) to 6/30 or 20% of natural numbers.

%C This is subset of natural numbers not divisible by 2, 3 or 5 (A007775).

%C Also A128464(n) and A128464(n)+2 interleaved, with a(n) = 1. - _Peter Bala_, Oct 28 2013

%H Gary W. Croft <a href="http://www.primesdemystified.com/twinprimes">Twin Primes Demystified</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).

%F G.f.: x*(1+10*x+2*x^2+4*x^3+2*x^4+10*x^5+x^6) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - _R. J. Mathar_, Jul 07 2015

%F From _Wesley Ivan Hurt_, Jul 22 2016: (Start)

%F a(n) = a(n-1) + a(n-6) - a(n-7) for n>7; a(n) = a(n-6) + 30 for n>6.

%F a(n) = (30*n - 15 - 6*cos(n*Pi/3) + 6*cos(2*n*Pi/3) + 9*cos(n*Pi) + 6*sqrt(3)*sin(n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/6.

%F a(6k) = 30k-1, a(6k-1) = 30k-11, a(6k-2) = 30k-13, a(6k-3) = 30k-17, a(6k-4) = 30k-19, a(6k-5) = 30k-29. (End)

%p A230462:=n->30*floor(n/6)+[1, 11, 13, 17, 19, 29][(n mod 6)+1]: seq(A230462(n), n=0..100); # _Wesley Ivan Hurt_, Jul 22 2016

%t LinearRecurrence[{1,0,0,0,0,1,-1},{1,11,13,17,19,29,31},60] (* _Harvey P. Dale_, Dec 01 2015 *)

%o (PARI) a(n)=n\6*30+[-1,1,11,13,17,19][n%6+1] \\ _Charles R Greathouse IV_, Oct 29 2013

%o (MAGMA) [n : n in [0..400] | n mod 30 in [1, 11, 13, 17, 19, 29]]; // _Wesley Ivan Hurt_, Jul 22 2016

%Y Cf. A001097, A007775, A128464, A132247.

%K nonn,easy

%O 1,2

%A _Gary Croft_, Oct 20 2013

%E New name (and initial term) from _Omar E. Pol_, Oct 27 2013

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Last modified August 17 23:58 EDT 2017. Contains 290682 sequences.