A158622 - OEIS

%I #17 Feb 26 2020 16:14:23

%S 7,13,7,31,43,19,73,91,37,133,157,61,211,241,91,307,343,127,421,463,

%T 169,553,601,217,703,757,271,871,931,331,1057,1123,397,1261,1333,469,

%U 1483,1561,547,1723,1807,631,1981,2071,721,2257,2353,817,2551,2653,919,2863

%N Numerator 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, ...

%C Is this the same as A046163? - _R. J. Mathar_, Mar 27 2009

%C Apparently a(n) = A130770(n) for 2 <= n <= 53. - _Georg Fischer_, Oct 24 2018

%H Harvey P. Dale, <a href="/A158622/b158622.txt">Table of n, a(n) for n = 2..1000</a>

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

%F A158620(n)/A158621(n) = 2(n^2+n+1)/(3n(n+1)). - _R. J. Mathar_, Mar 27 2009

%F Empirical g.f.: -x^2*(x^8 + x^7 + x^6 - 2*x^5 + 4*x^4 + 10*x^3 + 7*x^2 + 13*x + 7) / ((x-1)^3*(x^2 + x + 1)^3). - _Colin Barker_, May 09 2013

%e a(2) = 7 = numerator of (2^3-1)/2^3+1 = 7/9.

%e a(3) = 13 = numerator of ((2^3-1)*(3^3-1))/((2^3+1)*(3^3+1)) = (7 * 26)/ (9 * 28) = 182/252 = 13/18.

%e a(4) = 7 = = numerator 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.

%e a(5) = 31 = numerator 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 A158622 := proc(n) 2*(n^2+n+1)/3/n/(n+1) ; numer(%) ; end: seq(A158622(n),n=2..100) ; # _R. J. Mathar_, Mar 27 2009

%t Table[Product[k^3-1,{k,2,n}]/Product[k^3+1,{k,2,n}],{n,2,60}]//Numerator (* _Harvey P. Dale_, Feb 26 2020 *)

%Y Cf. A001093, A016921, A068601, A130770, A158620-A158621, A158623.

%K easy,nonn

%O 2,1

%A _Jonathan Vos Post_, Mar 23 2009

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