A168183 Numbers that are not multiples of 9.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Author

Reinhard Zumkeller, Nov 30 2009

Keywords

• nonn
• easy

Comments

It seems that, for any n >= 1, there exists no positive integer z such that digit_sum(z) = digit_sum(a(n)+z). - Max Lacoma, Sep 19 2019. Giovanni Resta: this follows immediately from the well-known fact that sod(x) == x (mod 9).

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Crossrefs

Complement of A008591.

Cf. A010887, A010888, A109012, A168182, A248375.

Programs

Haskell

a168183 n = a168183_list !! (n-1)
a168183_list = [1..8] ++ map (+ 9) a168183_list
-- Reinhard Zumkeller, Mar 04 2014

Magma

[n+Floor((n-1)/8) : n in [1..100]]; // Wesley Ivan Hurt, Sep 12 2015

Maple

A168183:=n->n+floor((n-1)/8): seq(A168183(n), n=1..100); # Wesley Ivan Hurt, Sep 12 2015

Mathematica

Select[Table[n,{n,200}],Mod[#,9]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011 *)
With[{nn=81},Complement[Range[nn],9Range[Floor[nn/9]]]] (* Harvey P. Dale, Sep 07 2011 *)

PARI

is(n)=!!(n%9) \\ Charles R Greathouse IV, Sep 02 2015

PARI

a(n)=(9*n-1)\8 \\ Charles R Greathouse IV, Sep 02 2015

Python

from gmpy2 import f_mod
[n for n in range(100) if f_mod(n,9)] # Bruno Berselli, Dec 05 2016

Formula

A168182(a(n)) = 1.

A010888(a(n)) = A010887(n-1).

A109012(a(n)) < 9.

From Wesley Ivan Hurt, Sep 12 2015: (Start)

a(n) = a(n-1) + a(n-8) - a(n-9), n>9.

a(n) = n + floor((n-1)/8). (End)

From Philippe Deléham, Dec 05 2016: (Start)

a(n) = 1 + A248375(n-1).

G.f.: x*(1-x^9)/((1-x)^2*(1-x^8)). (End)

E.g.f.: 1 + (1/8)*(-cos(x) + (-5+9*x)*cosh(x) - 2*cos(x/sqrt(2))*cosh(x/sqrt(2)) + sin(x) + (-4+9*x)*sinh(x) + 2*sin(x/sqrt(2))*(sqrt(2)*cosh(x/sqrt(2)) + sinh(x/sqrt(2)))). - Stefano Spezia, Sep 20 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(3) + 3*cosec(2*Pi/9) - 3*tan(Pi/18)) * Pi/27. - Amiram Eldar, May 11 2025