A144664 - OEIS

%I #35 Mar 24 2026 04:15:34

%S 9,2,8,7,8,6,9,3,5,7,9,9,5,5,2,4,5,3,7,5,1,4,6,9,9,1,5,6,5,2,8,5,2,3,

%T 5,1,9,3,2,0,1,0,1,5,0,3,7,5,3,0,4,1,1,8,2,0,1,0,2,8,2,6,5,1,4,8,7,2,

%U 0,0,7,3,7,9,9,1,6,0,2,2,3,8,8,2,7,4,1,5,5,1,8,1,0,8,4,1,9,2,7,8,2,5,1,0,5,9,7,2,6,2

%N Decimal expansion of Product_{n>=2} (n^5-1)/(n^5+1).

%H J. Borwein et al., <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.

%H Ryan Goulden, <a href="https://arxiv.org/abs/2602.06244">Closed-Form Evaluation of Arctanh Power Sums via Infinite Products</a>, arXiv preprint arXiv:2602.06244 [math.GM], 2026.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.

%F Equals exp(-2*Sum_{k>=2} arctanh(1/k^5)). - _Detlef Meya_, Mar 23 2026

%e 0.92878693579955245375146991...

%p evalf[120](exp(-2*sum(arctanh(1/k^5), k=2..infinity)));  # _Alois P. Heinz_, Mar 23 2026

%t p = 2*Gamma[2-(-1)^(1/5)] * Gamma[2+(-1)^(2/5)] * Gamma[2-(-1)^(3/5)] * Gamma[2+(-1)^(4/5)] / (Gamma[2+(-1)^(1/5)] * Gamma[2-(-1)^(2/5)] * Gamma[2+(-1)^(3/5)] * Gamma[2-(-1)^(4/5)]); RealDigits[Re[p], 10, 110][[1]] (* _Jean-François Alcover_, Feb 11 2013, updated Nov 18 2015 *)

%t RealDigits[Re[N[Product[(n^5 - 1)/(n^5 + 1), {n, 2, Infinity}], 110]]][[1]] (* _Bruno Berselli_, Apr 02 2013 *)

%t RealDigits[Exp[-2 NSum[ArcTanh[1/k^5], {k, 2, Infinity}, WorkingPrecision -> 120, NSumTerms -> 200, Method -> {"NIntegrate", "MaxRecursion" -> 18}]], 10, 110][[1]] (* _Detlef Meya_, Mar 23 2026 *)

%Y Cf. A090986, A255435.

%K nonn,cons

%O 0,1

%A _R. J. Mathar_, Feb 01 2009

%E More terms from _Jean-François Alcover_, Feb 11 2013