A047592 Numbers that are congruent to {1, 2, 3, 4, 5, 6, 7} mod 8.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Or, numbers that are not multiples of 8. - Benoit Cloitre, Jul 11 2009

More generally the sequence of numbers not divisible by some fixed integer m >= 2 is given by a(n, m) = n - 1 + floor((n+m-2)/(m-1)). - Benoit Cloitre, Jul 11 2009

Complement of A008590. - Reinhard Zumkeller, Nov 30 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Crossrefs

Cf. A008590, A168181, A207481.

Programs

Magma

[ n: n in [0..100] | n mod 8 in {1, 2, 3, 4, 5, 6, 7} ]; // Vincenzo Librandi, Dec 25 2010

Maple

A047592:=n->8*floor(n/7)+[1, 2, 3, 4, 5, 6, 7][(n mod 7)+1]: seq(A047592(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Mathematica

Complement[Range[88], 8Range[11]] (* Harvey P. Dale, Jan 22 2011 *)
CoefficientList[Series[(1 + x)*(1 + x^2)*(1 + x^4)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2013 *)

PARI

a(n)=n-1+floor((n+6)/7) \\ Benoit Cloitre, Jul 11 2009

Formula

a(n) = n - 1 + floor((n+6)/7). - Benoit Cloitre, Jul 11 2009

A168181(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009

From R. J. Mathar, Mar 08 2011: (Start)

a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.

G.f.: x*(1+x)*(1+x^2)*(1+x^4) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). (End)

a(n) = A207481(n) for n <= 70. - Reinhard Zumkeller, Feb 18 2012

From Wesley Ivan Hurt, Jul 20 2016: (Start)

a(n) = (56*n - 28 + (n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) - 6*((n+6) mod 7))/49.

a(7k) = 8k-1, a(7k-1) = 8k-2, a(7k-2) = 8k-3, a(7k-3) = 8k-4, a(7k-4) = 8k-5, a(7k-5) = 8k-6, a(7k-6) = 8k-7. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (4*sqrt(sqrt(2)+2) - 2*sqrt(2) - 1)*Pi/16. - Amiram Eldar, Dec 28 2021