A206547 Positive odd numbers relatively prime to 21.

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 131, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 209, 211

Offset: 1

Author

Wolfdieter Lang, Feb 10 2012

Keywords

• nonn
• easy

Comments

These are the positive integers not divisible by 2, 3, or 7.

Numbers coprime to 42. The asymptotic density of this sequence is 2/7. - Amiram Eldar, Oct 23 2020

Links

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

Crossrefs

Cf. A007775, A204458.

Programs

Mathematica

Select[Range[1,211,2],CoprimeQ[#,21]&] (* Harvey P. Dale, Jul 28 2020 *)

Formula

a(n) = a(n-12) + 42, n>=13.

a(n) = a(n-1) + a(n-12) - a(n-13), n>=13, with a(0)=-1.

a(n) = 2*n-1 + 2*sum(F21[j]*floor((n+(j-1))/12),j=1..12), with F21=[1,2,0,1,0,1,0,1,0,2,1,0], n>=1. For n=0 this becomes -1, but the following o.g.f. has a(0)=0 if it starts with x^0.

O.g.f.: x*(1+x^12+4*x*(1+x^10)+6*x^2*(1+x^8)+2*x^3*(1+x^6)+4*x^4*(1+x^4)+2*x^5*(1+x^2)+4*x^6)/((1-x^12)*(1-x)). The denominator could be factored into cyclotomic polynomials. Compare with the formula contribution from R. J. Mathar in A007775.