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%I #84 Dec 02 2023 12:03:36
%S 1,1,3,1,5,3,7,2,9,5,11,3,13,7,15,4,17,9,19,5,21,11,23,6,25,13,27,7,
%T 29,15,31,8,33,17,35,9,37,19,39,10,41,21,43,11,45,23,47,12,49,25,51,
%U 13,53,27,55,14,57,29,59,15,61,31,63,16,65,33,67,17,69,35,71,18,73,37,75,19
%N a(n) = n / gcd(n,4).
%C From _Peter Bala_, Feb 19 2019: (Start)
%C We make some general remarks about the sequence a(n) = numerator(n/(n + k)) = n/gcd(n,k) for k a fixed positive integer. The present sequence is the case k = 4. Several other cases are listed in the Crossrefs. In addition to being multiplicative these sequences are also strong divisibility sequences, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m).
%C By the multiplicativeness and strong divisibility property of the sequence a(n) it follows that if gcd(n, m) = 1 then a(a(n)*a(m) ) = a(a(n)) * a(a(m)), a(a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.
%C The sequence a(n) has the rational generating function Sum_{d divides k} f(d)*x^d/(1 - x^d)^2, where f(n) is the Dirichlet inverse of the Euler totient function A000010. f(n) is a multiplicative function defined on prime powers p^k by f(p^k) = 1 - p. See A023900. Cf. A181318. (End)
%H Harry J. Smith, <a href="/A060819/b060819.txt">Table of n, a(n) for n = 1..1000</a>
%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_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,-1).
%F G.f.: x*(1 +x +3*x^2 +x^3 +3*x^4 +x^5 +x^6)/(1 - x^4)^2.
%F a(n) = 2*a(n-4) - a(n-8).
%F a(n) = (n/16)*(11 - 5*(-1)^n - i^n - (-i)^n). - _Ralf Stephan_, Mar 15 2003
%F a(2*n+1) = a(4*n+2) = 2*n+1, a(4*n+4) = n+1. - _Ralf Stephan_, Jun 10 2005
%F Multiplicative with a(2^e) = 2^max(0, e-2), a(p^e) = p^e, p >= 3. - _Mitch Harris_, Jun 29 2005
%F a(n) = A167192(n+4,4). - _Reinhard Zumkeller_, Oct 30 2009
%F From _R. J. Mathar_, Apr 18 2011: (Start)
%F a(n) = A109045(n)/4.
%F Dirichlet g.f.: zeta(s-1)*(1-1/2^s-1/2^(2s)). (End)
%F a(n+4) - a(n) = A176895(n). - _Paul Curtz_, Apr 05 2011
%F a(n) = numerator(Sum_{k=1..n} 1/((k+1)*(k+2))). This summation has a closed form of 1/2 - 1/(n+2) and denominator of A145979(n). - _Gary Detlefs_, Sep 16 2011
%F a((2*n-1)*2^p) = ceiling(2^(p-2))*(2*n-1), p >= 0 and n >= 1. - _Johannes W. Meijer_, Feb 06 2013
%F a(n) = n / A109008(n). - _Reinhard Zumkeller_, Nov 25 2013
%F a(n) = denominator((2n-4)/n). - _Wesley Ivan Hurt_, Dec 22 2016
%F From _Peter Bala_, Feb 21 2019: (Start)
%F O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4), where F(x) = x/(1 - x)^2.
%F More generally, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) ), 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 operator to the o.g.f. for the sequence a(n) produces generating functions for the sequences ((n^m)*a(n))n>=1 for m in Z. Some examples are given below.
%F (End)
%F Sum_{k=1..n} a(k) ~ (11/32) * n^2. - _Amiram Eldar_, Nov 25 2022
%F E.g.f.: x*(8*cosh(x) + sin(x) + 3*sinh(x))/8. - _Stefano Spezia_, Dec 02 2023
%e From _Peter Bala_, Feb 21 2019: (Start)
%e Sum_{n >= 1} n*a(n)*x^n = G(x) - 2*G(x^2) - 4*G(x^4), where G(x) = x*(1 + x)/(1 - x)^3.
%e Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (1/2)*H(x^2) - (1/4)*H(x^4), where H(x) = x/(1 - x).
%e Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4^2)*L(x^4), where L(x) = Log(1/(1 - x)).
%e Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4). (End)
%p A060819 := n -> numer(1/2-1/(n+2)): seq(A060819(n),n=1..75); # _Gary Detlefs_, Sep 16 2011
%t f[n_]:= n/GCD[n, 4]; Array[f, 80]
%o (Sage) [lcm(n,4)/4 for n in (1..80)] # _Zerinvary Lajos_, Jun 07 2009
%o (PARI) { for (n=1, 1000, write("b060819.txt", n, " ", n / gcd(n, 4)); ) } \\ _Harry J. Smith_, Jul 12 2009
%o (Haskell)
%o a060819 n = n `div` a109008 n -- _Reinhard Zumkeller_, Nov 25 2013
%o (Magma) [n/GCD(n,4): n in [1..80]]; // _G. C. Greubel_, Sep 19 2018
%o (GAP) List([1..80],n->n/Gcd(n,4)); # _Muniru A Asiru_, Feb 20 2019
%Y Cf. A026741, A051176, A060791, A060789. Cf. Other sequences given by the formula numerator(n/(n + k)): A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).
%Y Cf. A000010, A023900, A061037, A061038, A109008, A145979, A167192, A176895, A181318, A220466.
%K nonn,mult,easy
%O 1,3
%A _Len Smiley_, Apr 30 2001
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