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

%I A230462 #73 Jul 29 2024 06:14:37
%S A230462 1,11,13,17,19,29,31,41,43,47,49,59,61,71,73,77,79,89,91,101,103,107,
%T A230462 109,119,121,131,133,137,139,149,151,161,163,167,169,179,181,191,193,
%U A230462 197,199,209,211,221,223,227,229,239,241,251,253,257,259,269
%N A230462 Numbers congruent to {1, 11, 13, 17, 19, or 29} mod 30.
%C A230462 Reduces sieving for all twin primes (A001097) except (3,5) and (5,7) to 6/30 or 20% of natural numbers.
%C A230462 This is subset of natural numbers not divisible by 2, 3 or 5 (A007775).
%C A230462 Also A128464(n) and A128464(n)+2 interleaved, with a(n) = 1. - _Peter Bala_, Oct 28 2013
%C A230462 a(2)..a(10) form a block of 9 primes {11, 13, 17, 19, 29, 31, 41, 43, 47}. Up to 3*10^10 there is only one such block which includes 11 primes: {18873497, 18873499, 18873509, 18873511, 18873521, 18873523, 18873527, 18873529, 18873539, 18873541, 18873551}. Do larger such blocks exist? (None found up to 10^11.) - _Mikk Heidemaa_, Dec 22 2017
%H A230462 Iain Fox, <a href="/A230462/b230462.txt">Table of n, a(n) for n = 1..10000</a>
%H A230462 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).
%F A230462 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 A230462 From _Wesley Ivan Hurt_, Jul 22 2016: (Start)
%F A230462 a(n) = a(n-1) + a(n-6) - a(n-7) for n>7; a(n) = a(n-6) + 30 for n>6.
%F A230462 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 A230462 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)
%F A230462 a(n) = 5*n + ceiling(7/79 - ((((14654/4883)^n mod 6) mod 5) + n mod 3 + 1) mod 7). - _Mikk Heidemaa_, Dec 13 2017
%F A230462 a(n + 6) = a(n) + 30. - _David A. Corneth_, Jan 15 2018
%F A230462 Sum_{n>=1} (-1)^(n+1)/a(n) = (cot(Pi/30) - sec(2*Pi/15) * tan(Pi/15)) * Pi/30. - _Amiram Eldar_, Jul 29 2024
%p A230462 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 A230462 LinearRecurrence[{1,0,0,0,0,1,-1},{1,11,13,17,19,29,31},60] (* _Harvey P. Dale_, Dec 01 2015 *)
%t A230462 ParallelCombine[Select[#, MemberQ[{1, 11, 13, 17, 19, 29}, Mod[#, 30]] &] &, Range[10^4]] (* _Mikk Heidemaa_, Dec 12 2017 *)
%t A230462 CoefficientList[ Series[(1 + 10x + 2x^2 + 4x^3 + 2x^4 + 10x^5 + x^6)/((-1 + x)^2 (1 + x + x^2 + x^3 + x^4 + x^5)), {x, 0, 60}], x] (* _Robert G. Wilson v_, Jan 09 2018 *)
%o A230462 (PARI) a(n)=n\6*30+[-1,1,11,13,17,19][n%6+1] \\ _Charles R Greathouse IV_, Oct 29 2013
%o A230462 (PARI) first(n) = Vec(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) + O(x^(n+1))) \\ _Iain Fox_, Dec 29 2017
%o A230462 (Magma) [n : n in [0..400] | n mod 30 in [1, 11, 13, 17, 19, 29]]; // _Wesley Ivan Hurt_, Jul 22 2016
%Y A230462 Cf. A001097, A007775, A128464, A132247.
%K A230462 nonn,easy
%O A230462 1,2
%A A230462 _Gary Croft_, Oct 20 2013
%E A230462 New name and initial term from _Omar E. Pol_, Oct 27 2013