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Denominator of the reduced fraction A158620(n)/A158621(n).
4

%I #9 Feb 01 2019 20:43:57

%S 9,18,10,45,63,28,108,135,55,198,234,91,315,360,136,459,513,190,630,

%T 693,253,828,900,325,1053,1134,406,1305,1395,496,1584,1683,595,1890,

%U 1998,703,2223,2340,820,2583,2709,946,2970,3105,1081,3384,3528,1225,3825

%N Denominator of the reduced fraction A158620(n)/A158621(n).

%C A158620(n) = Product_{k=2..n} (k^3-1). A158621(n) = Product_{k=2..n} (k^3+1). A158622(n) is the numerator of the reduced fraction A158620(n)/A158621(n). A158623(n) is the denominator of the reduced fraction A158620(n)/A158621(n). The reduced fractions are 7/9, 13/18, 7/10, 31/45, 43/63, 19/28, 73/108, 91/135, 37/55, 133/198, ...

%F Denominator of (Product_{k=2..n} (k^3-1)) / Product_{k=2..n} (k^3+1) = denominator of Product_{k=2..n} A068601(k)/A001093(k).

%F A158620(n)/A158621(n) = 2(n^2+n+1)/(3n(n+1)). Conjecture: a(n) = 3a(n-3) - 3a(n-6) + a(n-9), so trisections are A152996, A060544 and 3*A081266. - _R. J. Mathar_, Mar 27 2009

%F Empirical g.f.: -x^2*(x^8 - 2*x^5 + 9*x^4 + 18*x^3 + 10*x^2 + 18*x + 9) / ((x-1)^3*(x^2 + x + 1)^3). - _Colin Barker_, May 09 2013

%e a(2) = 9 = denominator of (2^3-1)/2^3+1 = 7/9. a(3) = 18 = denominator of ((2^3-1)*(3^3-1))/((2^3+1)*(3^3+1)) = (7 * 26)/ (9 * 28) = 182/252 = 13/18. a(4) = 10 = denominator of ((2^3-1)*(3^3-1)*(4^3-1))/((2^3+1)*(3^3+1)*(4^3+1)) = (7 * 26 * 63)/(9 * 28 * 65) = 11466/16380 = 7/10. a(5) = 45 = denominator of ((2^3-1)(3^3-1)(4^3-1)(5^3-1))/((2^3+1)(3^3+1)(4^3+1)(5^3+1)) = 1421784/2063880 = 31/45.

%p A158623 := proc(n) 2*(n^2+n+1)/3/n/(n+1) ; denom(%) ; end: seq(A158623(n),n=2..100) ; # _R. J. Mathar_, Mar 27 2009

%Y Cf. A001093, A016921, A068601, A158620-A158622.

%K easy,nonn

%O 2,1

%A _Jonathan Vos Post_, Mar 23 2009

%E More terms from _R. J. Mathar_, Mar 27 2009