A204457 Odd numbers not divisible by 13.

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 93, 95, 97, 99, 101, 103, 105, 107, 109

Offset

1,2

Comments

For the general case of odd numbers not divisible by primes see a comment on A204454, where the o.g.f.s and the formulas in terms of floor functions are given.

The numerator polynomial of the o.g.f. given in the formula section has coefficients 1,2,2,2,2,2,4,2,2,2,2,2,1, see row no. 6 of A204456. The first seven numbers are the first differences of the sequence, starting with a(0)=0. The other numbers are obtained by mirroring around the center.

Numbers coprime to 26. The asymptotic density of this sequence is 6/13. - Amiram Eldar, Oct 20 2020

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Formula

O.g.f.: x*(1 + 2*(x+x^6)*(1+x+x^2+x^3+x^4) + 4*x^6 + x^12)/((1-x^12)*(1-x)). The denominator can be factored.

a(n) = 2*n-1 + 2*floor((n+5)/12) = 2*n+1 + 2*floor((n-7)/12), n>=1. Note that this is -1 for n=0, but the o.g.f. starting with x^0 has a(0)=0.

Mathematica

Select[Range[1, 111, 2], !Divisible[#, 13]&] (* or *) With[{nn=111}, Complement[ Range[1, nn, 2], 13*Range[Floor[nn/13]]]] (* Harvey P. Dale, Jul 23 2013 *)

Prog

(Haskell)
a204457 n = a204457_list !! (n-1)
a204457_list = [x | x <- [1, 3 ..], mod x 13 > 0]
-- Reinhard Zumkeller, Feb 08 2012
(PARI) a(n) = 2*n-1+(n+5)\12*2 \\ Charles R Greathouse IV, Feb 08 2012

Crossrefs

Cf. A204454 and cross-references there; A204458.

Sequence in context: A165704 A275852 A300914 * A283800 A043325 A023692

Adjacent sequences: A204454 A204455 A204456 * A204458 A204459 A204460

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 07 2012

Status

approved