A106609 - OEIS

%I #90 Nov 25 2022 13:18:07

%S 0,1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,2,17,9,19,5,21,11,23,3,25,13,27,7,

%T 29,15,31,4,33,17,35,9,37,19,39,5,41,21,43,11,45,23,47,6,49,25,51,13,

%U 53,27,55,7,57,29,59,15,61,31,63,8,65,33,67,17,69,35,71,9,73,37,75,19,77,39,79

%N Numerator of n/(n+8).

%C The graph of this sequence is made up of four linear functions: a(n_odd)=n, a(n=2+4i)=n/2, a(4+8i)=n/4, a(8i)=n/8. - _Zak Seidov_, Oct 30 2006. [In general, f(n) = numerator of n/(n+m) consists of linear functions n/d_i, where d_i are divisors of m (including 1 and m).]

%C a(n+2), n>=0, is the denominator of the harmonic mean H(n,2) = 4*n/(n+2). a(n+2) = (n+2)/gcd(n+2,8). a(n+5) = A227042(n+2, 2), n >= 0. - _Wolfdieter Lang_, Jul 04 2013

%C The sequence p(n) = a(n-4), n>=1, with a(-3) = a(3) = 3, a(-2) = a(2) = 1 and a(-1) = a(1) = 1, appears in the problem of writing 2*sin(2*Pi/n) as an integer in the algebraic number field Q(rho(q(n))), where rho(k) = 2*cos(Pi/k) and q(n) = A225975(n). One has 2*sin(2*Pi/n) = R(p(n), x) modulo C(q(n), x), with x = rho(q(n)) and the integer polynomials R and C given in A127672 and A187360, respectively. See a comment on A225975. - _Wolfdieter Lang_, Dec 04 2013

%C A204455(n) divides a(n) for n>=1. - _Alexander R. Povolotsky_, Apr 06 2015

%C A multiplicative sequence. Also, 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 20 2019

%H N. J. A. Sloane, <a href="/A106609/b106609.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

%F a(n) = 2*a(n-8) - a(n-16).

%F G.f.: x* (x^2-x+1) * (x^12 +2*x^11 +4*x^10 +3*x^9 +4*x^8 +4*x^7 +7*x^6 +4*x^5 +4*x^4 +3*x^3 +4*x^2 +2*x +1) / ( (x-1)^2 *(x+1)^2 *(x^2+1)^2 *(x^4+1)^2 ). - _R. J. Mathar_, Dec 02 2010

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

%F a(n) = A109049(n)/8.

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

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

%F a(n) = n/gcd(n,8), n >= 0. See the harmonic mean comment above. - _Wolfdieter Lang_, Jul 04 2013

%F a(n) = n if n is odd; for n == 0 (mod 8) it is n/8, for n == 2 or 6 (mod 8) it is n/2 and for n == 4 (mod 8) it is n/4. - _Wolfdieter Lang_, Dec 04 2013

%F From _Peter Bala_, Feb 20 2019: (Start)

%F O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4) - F(x^8), 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 - 2^m)*( F(m,x^2) + F(m,x^4) + F(m,x^8) ), 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 Sum_{n >= 1} (1/n)*a(n)*x^n = G(x) - (1/2)*G(x^2) - (1/4)*G(x^4) - (1/8)*G(x^8), where G(x) = x/(1 - x).

%F Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4)^2*L(x^4) - (1/8)^2*L(x^8), where L(x) = Log(1/(1 - x)).

%F Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4) + (1/2)*L(x^8). (End)

%F Sum_{k=1..n} a(k) ~ (43/128) * n^2. - _Amiram Eldar_, Nov 25 2022

%p a := n -> iquo(n, [8, 1, 2, 1, 4, 1, 2, 1][1 + modp(n, 8)]):

%p seq(a(n), n=0..79); # using _Wolfdieter Lang_'s formula, _Peter Luschny_, Feb 22 2019

%t f[n_]:=Numerator[n/(n+8)];Array[f,100,0] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2011 *)

%t LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,1,1,3,1,5,3,7,1,9,5,11,3,13,7,15},100] (* _Harvey P. Dale_, Sep 27 2019 *)

%o (Sage) [lcm(n,8)/8 for n in range(0, 100)] # _Zerinvary Lajos_, Jun 09 2009

%o (Magma) [Numerator(n/(n+8)): n in [0..100]]; // _Vincenzo Librandi_, Apr 18 2011

%o (PARI) vector(100, n, n--; numerator(n/(n+8))) \\ _G. C. Greubel_, Feb 19 2019

%o (GAP) List([0..80],n->NumeratorRat(n/(n+8))); # _Muniru A Asiru_, Feb 19 2019

%Y Cf. A109049, A204455, A225975, A227042 (second column, starting with a(5)).

%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,frac,mult,easy

%O 0,4

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