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%I #41 Sep 08 2023 07:32:49
%S 0,1,2,3,4,5,6,7,8,9,10,11,12,1,14,15,16,17,18,19,20,21,22,23,24,25,2,
%T 27,28,29,30,31,32,33,34,35,36,37,38,3,40,41,42,43,44,45,46,47,48,49,
%U 50,51,4,53,54,55,56,57,58,59,60,61,62,63,64,5,66,67,68,69,70,71,72,73,74,75
%N a(n) = numerator of n/(n+13).
%C In general, the numerators of n/(n+p) for prime p and n >= 0, form a sequence with the g.f.: x/(1-x)^2 - (p-1)*x^p/(1-x^p)^2. - _Paul D. Hanna_, Jul 27 2005
%C a(n) <> n iff n = 13 * k, in this case, a(n) = k. - _Bernard Schott_, Feb 19 2019
%H G. C. Greubel, <a href="/A106614/b106614.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_26">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,-1).
%F G.f.: x/(1-x)^2 - 12*x^13/(1-x^13)^2. - _Paul D. Hanna_, Jul 27 2005
%F Dirichlet g.f.: zeta(s-1)*(1-12/13^s). - _R. J. Mathar_, Apr 18 2011
%F a(n) = 2*a(n-13) - a(n-26). - _G. C. Greubel_, Feb 19 2019
%F From _Amiram Eldar_, Nov 25 2022: (Start)
%F Multiplicative with a(13^e) = 13^(e-1), and a(p^e) = p^e if p != 13.
%F Sum_{k=1..n} a(k) ~ (157/338) * n^2. (End)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 25*log(2)/13. - _Amiram Eldar_, Sep 08 2023
%p seq(numer(n/(n+13)),n=0..80); # _Muniru A Asiru_, Feb 19 2019
%t f[n_]:=Numerator[n/(n+13)];Array[f,100,0] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2011 *)
%o (Sage) [lcm(n,13)/13for n in range(0, 100)] # _Zerinvary Lajos_, Jun 09 2009
%o (Magma) [Numerator(n/(n+13)): n in [0..100]]; // _Vincenzo Librandi_, Apr 18 2011
%o (PARI) vector(100, n, n--; numerator(n/(n+13))) \\ _G. C. Greubel_, Feb 19 2019
%o (PARI) a(n)=if(n%13,n,n/13) \\ _Charles R Greathouse IV_, Jan 24 2022
%o (GAP) List([0..80],n->NumeratorRat(n/(n+13))); # _Muniru A Asiru_, Feb 19 2019
%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), A106615 thru A106621 (k = 14 thru 20).
%K nonn,frac,mult,easy
%O 0,3
%A _N. J. A. Sloane_, May 15 2005
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