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

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

Offset: 1

Author

Gary Croft, Oct 20 2013

Keywords

• nonn
• easy

Comments

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).

Also A128464(n) and A128464(n)+2 interleaved, with a(n) = 1. - Peter Bala, Oct 28 2013

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

Links

• Iain Fox, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Crossrefs

Cf. A001097, A007775, A128464, A132247.

Programs

Magma

[n : n in [0..400] | n mod 30 in [1, 11, 13, 17, 19, 29]]; // Wesley Ivan Hurt, Jul 22 2016

Maple

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

Mathematica

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 *)

PARI

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

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

Formula

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

From Wesley Ivan Hurt, Jul 22 2016: (Start)

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

a(n + 6) = a(n) + 30. - David A. Corneth, Jan 15 2018

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

Extensions

New name and initial term from Omar E. Pol, Oct 27 2013