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%I #57 Nov 25 2022 13:18:23
%S 0,1,1,1,1,5,1,7,2,3,5,11,1,13,7,5,4,17,3,19,5,7,11,23,2,25,13,9,7,29,
%T 5,31,8,11,17,35,3,37,19,13,10,41,7,43,11,15,23,47,4,49,25,17,13,53,9,
%U 55,14,19,29,59,5,61,31,21,16,65,11,67,17,23,35,71
%N Numerator of n/12.
%C Or, numerator of n/(n+12).
%C Arises in the equal-tempered musical scale - see Goldstein (1977), Table 1. - _N. J. A. Sloane_, Aug 29 2018
%C A strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - _Peter Bala_, Feb 24 2019
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 269.
%H Nathaniel Johnston, <a href="/A051724/b051724.txt">Table of n, a(n) for n = 0..10000</a>
%H Peter Bala, <a href="/A306367/a306367.pdf">A note on the sequence of numerators of a rational function</a>, 2019.
%H A. A. Goldstein, <a href="https://www.jstor.org/stable/2029622">Optimal temperament</a>, SIAM Review 19.3 (1977): 554-562.
%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-1).
%F From _David W. Wilson_, Jun 12 2005: (Start)
%F a(n) = n/gcd(n, 12).
%F Multiplicative with a(2^e) = 2^max(0, e-2), a(3^e) = 3^max(0,e-1), a(p^e) = p^e otherwise. (End)
%F From _R. J. Mathar_, Apr 18 2011: (Start)
%F a(n) = A109053(n)/12.
%F Dirichlet g.f.: zeta(s-1)*(1 - 2/3^s - 1/2^s + 2/6^s - 1/4^s + 2/12^s). (End)
%F From _Peter Bala_, Feb 24 2019: (Start)
%F a(n) = n/gcd(n,12) is a quasi-polynomial in n since gcd(n,12) is a purely periodic sequence of period 12.
%F O.g.f.: F(x) - F(x^2) - 2*F(x^3) - F(x^4) + 2*F(x^6) + 2*F(x^12), where F(x) = x/(1 - x)^2.
%F O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 12} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (1/2)*log(1/(1 - x^2)) + (2/3)*log(1/(1 - x^3)) + (2/4)*log(1/(1 - x^4)) + (2/6)*log(1/(1 - x^6)) + (4/12)*log(1/(1 - x^12)), where phi(n) denotes the Euler totient function A000010. (End)
%F Sum_{k=1..n} a(k) ~ (77/288) * n^2. - _Amiram Eldar_, Nov 25 2022
%p seq(numer(n/12),n=0..100); # _Nathaniel Johnston_, Apr 18 2011
%t f[n_]:=Numerator[n/(n+12)];Array[f,100,0] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2011*)
%o (Sage) [lcm(n,12)/12 for n in range(0, 72)] # _Zerinvary Lajos_, Jun 09 2009
%o (Magma) [Numerator(n/12): n in [0..100]]; // _Vincenzo Librandi_, Apr 18 2011
%o (PARI) a(n) = numerator(n/12); \\ _Michel Marcus_, Aug 19 2018
%o (GAP) List([0..80],n->NumeratorRat(n/12)); # _Muniru A Asiru_, Feb 24 2019
%Y Cf. A000010, A109053.
%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), A106614 thru A106621 (k = 13 thru 20).
%K nonn,easy,frac,mult
%O 0,6
%A _N. J. A. Sloane_
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