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%I #48 Sep 08 2023 07:32:57
%S 0,1,2,3,4,5,6,1,8,9,10,11,12,13,2,15,16,17,18,19,20,3,22,23,24,25,26,
%T 27,4,29,30,31,32,33,34,5,36,37,38,39,40,41,6,43,44,45,46,47,48,7,50,
%U 51,52,53,54,55,8,57,58,59,60,61,62,9,64,65,66,67,68,69,10,71,72,73,74,75,76
%N a(n) = numerator of n/(n+7).
%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 Generalizing the above comment of Hanna, the numerators of n/(n + k) for a fixed positive integer k and n >= 0 form a sequence with the g.f.: 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. - _Peter Bala_, Feb 17 2019
%C a(n) <> n iff n = 7 * k, in this case, a(n) = k. - _Bernard Schott_, Feb 19 2019
%H G. C. Greubel, <a href="/A106608/b106608.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).
%F G.f.: x/(1-x)^2 - 6*x^7/(1-x^7)^2. - _Paul D. Hanna_, Jul 27 2005
%F From _R. J. Mathar_, Apr 18 2011: (Start)
%F a(n) = A109048(n)/7.
%F Dirichlet g.f.: zeta(s-1)*(1-6/7^s).
%F Multiplicative with a(p^e) = p^e if p<>7, a(7^e) = 7^(e-1) if e > 0. (End)
%F Sum_{k=1..n} a(k) ~ (43/98) * n^2. - _Amiram Eldar_, Nov 25 2022
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 13*log(2)/7. - _Amiram Eldar_, Sep 08 2023
%p seq(numer(n/(n+7)),n=0..80); # _Muniru A Asiru_, Feb 19 2019
%t f[n_]:=Numerator[n/(n+7)];Array[f,100,0] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2011 *)
%o (Sage) [lcm(n,7)/7 for n in range(0, 100)] # _Zerinvary Lajos_, Jun 09 2009
%o (Magma) [Numerator(n/(n+7)): n in [0..100]]; // _Vincenzo Librandi_, Apr 18 2011
%o (PARI) vector(100, n, n--; numerator(n/(n+7))) \\ _G. C. Greubel_, Feb 19 2019
%o (GAP) List([0..80],n->NumeratorRat(n/(n+7))); # _Muniru A Asiru_, Feb 19 2019
%Y Cf. A000010, A109048, A023900.
%Y Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106609 thru A106612 (k = 8 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
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