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Decimal expansion of constant K, the unique solution > 1 to 2*Pi*log(k) == Pi*(1 - 1/k).
1

%I #17 Feb 16 2025 08:32:55

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

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

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

%N Decimal expansion of constant K, the unique solution > 1 to 2*Pi*log(k) == Pi*(1 - 1/k).

%C Decimal expansion of the number K > 1 such that the surface area equals the volume of Gabriel's horn from x=1 to x=K, where x is the radial (central) axis and Gabriel's horn is a function y=1/x rotated about the x-axis.

%H L. Bayon, P. Fortuny Ayuso, J. M. Grau, A. M. Oller-Marcen, and M.M. Ruiz, <a href="https://arxiv.org/abs/1706.07185">The Best-or-Worst and the Postdoc problems</a>, arXiv:1706.07185 [math.PR], 2017.

%H L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, and P. M. Suarez, <a href="http://arxiv.org/abs/1603.03928">A variant of the Secretary Problem: the Best or the Worst</a>, arXiv preprint arXiv:1603.03928 [math.PR], 2016.

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

%F -1/(2*ProductLog(-1, -1/(2*sqrt(e))))

%e 0.2846681370408384616802256767697191309865026705850454315169314709776689439...

%t RealDigits[ -1/(2*ProductLog[ -1, -1/(2*Sqrt[E])]), 10, 128][[1]]

%K cons,nonn,changed

%O 0,1

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 23 2004