This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A013667 #49 Aug 05 2025 14:39:28
%S A013667 1,0,0,2,0,0,8,3,9,2,8,2,6,0,8,2,2,1,4,4,1,7,8,5,2,7,6,9,2,3,2,4,1,2,
%T A013667 0,6,0,4,8,5,6,0,5,8,5,1,3,9,4,8,8,8,7,5,6,5,4,8,5,9,6,6,1,5,9,0,9,7,
%U A013667 8,5,0,5,3,3,9,0,2,5,8,3,9,8,9,5,0,3,9,3,0,6,9,1,2,7,1,6,9,5,8
%N A013667 Decimal expansion of zeta(9).
%D A013667 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
%H A013667 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A013667 Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/zeta9.txt">Zeta(9)=sum(1/n^9, n=1..infinity); to 20000 digits</a>
%H A013667 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap101.html">Zeta(9) or sum(1/n**9, n=1..infinity);</a>
%F A013667 From _Peter Bala_, Dec 04 2013: (Start)
%F A013667 Definition: zeta(9) = Sum_{n >= 1} 1/n^9.
%F A013667 zeta(9) = 2^9/(2^9 - 1)*( Sum_{n even} n^7*p(n)*p(1/n)/(n^2 - 1)^10 ), where p(n) = n^4 + 10*n^2 + 5. See A013663, A013671 and A013675. (End)
%F A013667 zeta(9) = Sum_{n >= 1} (A010052(n)/n^(9/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(9/2) ). - _Mikael Aaltonen_, Feb 22 2015
%F A013667 zeta(9) = Product_{k>=1} 1/(1 - 1/prime(k)^9). - _Vaclav Kotesovec_, May 02 2020
%F A013667 From _Peter Bala_, Apr 27 2025: (Start)
%F A013667 zeta(9) = 1/9! * Integral_{x >= 0} x^9 * exp(x)/(exp(x) - 1)^2 dx = 2^9/(2^9 - 1) * 1/9! * Integral_{x >= 0} x^9 * exp(x)/(exp(x) + 1)^2 dx.
%F A013667 zeta(9) = 1/10! * Integral_{x >= 0} x^10 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3^5 * 5^3 * 7 * 17) * Integral_{x >= 0} x^10 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
%e A013667 1.0020083928260822...
%p A013667 evalf(Zeta(9)) ; # _R. J. Mathar_, Oct 16 2015
%t A013667 RealDigits[Zeta[9],10,100][[1]] (* _Harvey P. Dale_, Aug 27 2014 *)
%Y A013667 Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
%Y A013667 Cf. A023876, A023877.
%K A013667 nonn,cons
%O A013667 1,4
%A A013667 _N. J. A. Sloane_