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

Author

Wolfdieter Lang, Feb 07 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A204454 and cross-references there; A204458.

Programs

Haskell

a204457 n = a204457_list !! (n-1)
a204457_list = [x | x <- [1, 3 ..], mod x 13 > 0]
-- Reinhard Zumkeller, Feb 08 2012

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 *)

PARI

a(n) = 2*n-1+(n+5)\12*2 \\ Charles R Greathouse IV, Feb 08 2012

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.