%I #63 Nov 25 2022 13:17:58

%S 0,1,2,1,4,5,2,7,8,1,10,11,4,13,14,5,16,17,2,19,20,7,22,23,8,25,26,3,

%T 28,29,10,31,32,11,34,35,4,37,38,13,40,41,14,43,44,5,46,47,16,49,50,

%U 17,52,53,6,55,56,19,58,59,20,61,62,7,64,65,22,67,68,23,70,71,8,73,74,25,76,77

%N Numerator of n/(n+9).

%C Apart from 0, also numerator of Sum_{i=1..n} (1/((i+2)*(i+3))) = n/(3n+9). - _Bruno Berselli_, Nov 07 2012

%C In addition to being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - _Peter Bala_, Feb 21 2019

%D Raffaello Giusti, editore, Supplemento al Periodico di Matematica (Livorno), Jul 1902, p. 138 (Problem 421, case k=3).

%H Indranil Ghosh, <a href="/A106610/b106610.txt">Table of n, a(n) for n = 0..20000</a> (terms 0..1000 from Vincenzo Librandi)

%H Peter Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function</a>, 2019.

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0, 0,-1).

%F From _R. J. Mathar_, Apr 18 2011: (Start)

%F a(n) = A109050(n)/9.

%F Dirichlet g.f. zeta(s-1)*(1-2/3^s-2/9^s).

%F Multiplicative with a(3^e) = 3^max(0,e-2), a(p^e) = p^e if p<>3. (End)

%F a(n) = 2*a(n-9) - a(n-18). - _G. C. Greubel_, Feb 19 2019

%F From _Peter Bala_, Feb 21 2019: (Start)

%F a(n) = n/gcd(n,9), where gcd(n,9) = [1, 1, 3, 1, 1, 3, 1, 1, 9, ...] is a periodic sequence of period 9: a(n) is thus quasi_polynomial in n.

%F O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - 2*F(x^3) - 2*F(x^9), where F(x) = x/(1 - x)^2.

%F More generally, for m >= 1, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 3^m)*( F(m,x^3) + F(m,x^9) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.

%F Repeatedly applying the Euler operator x*d/dx or its inverse to the o.g.f. for the sequence produces generating functions for the sequences n^m*a(n), m in Z. Some examples are given below. (End)

%F Sum_{k=1..n} a(k) ~ (61/162) * n^2. - _Amiram Eldar_, Nov 25 2022

%e For n = 12, n/(n+9) = 12/21 = 4/7. So, a(12) = 4. - _Indranil Ghosh_, Jan 31 2017

%e From _Peter Bala_, Feb 21 2019: (Start)

%e Sum_{n >= 1} n*a(n)*x^n = G(x) - (2*3)*G(x^3) - (2*9)*G(x^9) , where G(x) = x*(1 + x)/(1 - x)^3.

%e Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (2/3)*H(x^3) - (2/9)*H(x^9), where H(x) = x/(1 - x).

%e Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (2/3^2)*L(x^3) - (2/9^2)*L(x^9), where L(x) = Log(1/(1 - x)).

%e Sum_{n >= 1} (1/a(n))*x^n = L(x) + (2/3)*L(x^3) + (2/3)*L(x^9). (End)

%p seq(numer(n/(n+9)),n=0..80); # _Muniru A Asiru_, Feb 19 2019

%t f[n_]:=Numerator[n/(n+9)]; Array[f,100,0] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2011*)

%o (Sage) [lcm(n,9)/9for n in range(0, 100)] # _Zerinvary Lajos_, Jun 09 2009

%o (Magma) [Numerator(n/(n+9)): n in [0..100]]; // _Vincenzo Librandi_, Apr 18 2011

%o (PARI) vector(100, n, n--; numerator(n/(n+9))) \\ _G. C. Greubel_, Feb 19 2019

%o (GAP) List([0..80],n->NumeratorRat(n/(n+9))); # _Muniru A Asiru_, Feb 19 2019

%Y Cf. A008292, A109050.

%Y Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

%K nonn,easy,frac,mult

%O 0,3

%A _N. J. A. Sloane_, May 15 2005