A229973 Numbers coprime to 39.

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 56, 58, 59, 61, 62, 64, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 92, 94, 95, 97, 98, 100, 101, 103, 106

Offset: 1

Author

Gary Detlefs, Oct 04 2013

Keywords

• nonn
• easy

Comments

Numbers not divisible by 3 or 13.

For n from 1 to 24, a(n) mod 39-n - floor(11*n/25)-2*floor(n/8) has a period of 24, consisting of all zeros except a -2 at indices 8, 16, and 24.

The asymptotic density of this sequence is 8/13. - Amiram Eldar, Oct 23 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1,-1,2,-1).

Crossrefs

Cf. A160545, A229829, A229968.

Programs

Maple

for n from 1 to 50 do if n mod 3<>0 and n mod 13<>0 then print(n) fi od

Mathematica

CoefficientList[Series[(x^22 + x^20 + x^18 + x^16 + 2 x^14 - x^12 + 3 x^11 - x^10 + 2 x^8 + x^6 + x^4 + x^2 + 1)/((x - 1)^2 (x + 1) (x^2 - x + 1) (x^2 + 1) (x^4 - x^2 + 1) (x^4 + 1) (x^8 - x^4 + 1)), {x, 0, 80}], x] (* Vincenzo Librandi, Oct 08 2013 *)
Select[Range[100], CoprimeQ[39, #] &] (* Amiram Eldar, Oct 23 2020 *)

Formula

a(n+24) = a(n) + 39.

a(n) = 39*floor((n-1)/24) + f(n) + floor(11*f(n)/25) + 2*floor(f(n)/8) - 2*floor(((n-1)mod 8)/7) + 40*floor(f(n-1)/23), where f(n) = n mod 24.

G.f.: x*(x^22+x^20+x^18+x^16+2*x^14-x^12+3*x^11-x^10+2*x^8+x^6+x^4+x^2+1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)*(x^4+1)*(x^8-x^4+1)). - Colin Barker, Oct 07 2013

Extensions

More terms from Colin Barker, Oct 07 2013

a(34) corrected by Vincenzo Librandi, Oct 08 2013