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A230462
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Numbers congruent to {1, 11, 13, 17, 19, or 29} mod 30.
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2
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1, 11, 13, 17, 19, 29, 31, 41, 43, 47, 49, 59, 61, 71, 73, 77, 79, 89, 91, 101, 103, 107, 109, 119, 121, 131, 133, 137, 139, 149, 151, 161, 163, 167, 169, 179, 181, 191, 193, 197, 199, 209, 211, 221, 223, 227, 229, 239, 241, 251, 253, 257, 259, 269
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OFFSET
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1,2
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COMMENTS
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Reduces sieving for all twin primes (A001097) except (3,5) and (5,7) to 6/30 or 20% of natural numbers.
This is subset of natural numbers not divisible by 2, 3 or 5 (A007775).
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
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LINKS
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FORMULA
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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
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7; a(n) = a(n-6) + 30 for n>6.
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.
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)
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
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MAPLE
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MATHEMATICA
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LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {1, 11, 13, 17, 19, 29, 31}, 60] (* Harvey P. Dale, Dec 01 2015 *)
ParallelCombine[Select[#, MemberQ[{1, 11, 13, 17, 19, 29}, Mod[#, 30]] &] &, Range[10^4]] (* Mikk Heidemaa, Dec 12 2017 *)
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 *)
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PROG
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(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
(Magma) [n : n in [0..400] | n mod 30 in [1, 11, 13, 17, 19, 29]]; // Wesley Ivan Hurt, Jul 22 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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New name and initial term from Omar E. Pol, Oct 27 2013
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STATUS
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approved
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