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Decimal expansion of 1 / (1 - gamma - log(3/2)) - 54, where gamma is the Euler-Mascheroni constant.
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%I #23 Sep 08 2022 08:45:31

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

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

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

%N Decimal expansion of 1 / (1 - gamma - log(3/2)) - 54, where gamma is the Euler-Mascheroni constant.

%C Continued fraction expansion is given in A131918.

%H G. C. Greubel, <a href="/A131917/b131917.txt">Table of n, a(n) for n = 1..10000</a>

%H Mark B. Villarino, <a href="http://arXiv.org/abs/0707.3950">Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number</a>, arXiv:0707.3950 [math.CA], 2007. Constant occurs in Theorem 7 (DeTemple-Wang), formula (1.14), page 6.

%F Equals (54 log(3/2) + 54 gamma - 53)/(1 - log(3/2) - gamma) = 1 / (1 - gamma - log(3/2)) - 54, where _Martin Fuller_ simplifies the constant which Villarino showed was implicitly given by DeTemple and Wang.

%e 3.739297519451184207366332869686151725662636854564...

%t RealDigits[1/(1 - EulerGamma - Log[3/2]) - 54, 10, 100][[1]] (* _G. C. Greubel_, Aug 29 2018 *)

%o (PARI) 1/(1 - Euler - log(3/2)) - 54 \\ _Michel Marcus_, Mar 11 2013

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 1/(1 - EulerGamma(R) - Log(3/2)) - 54; // _G. C. Greubel_, Aug 29 2018

%Y Cf. A001008, A001620, A131915, A131916, A131918.

%K cons,easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Jul 27 2007