A241005 - OEIS

%I #24 Jan 17 2020 05:27:46

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

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

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

%N Decimal expansion of gamma', the analog of Euler's constant when 1/x is replaced by 1/(x*log(x)).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.3 Generalized Euler Constants, p. 32.

%H Harold G. Diamond, <a href="http://projecteuclid.org/euclid.pjm/1102710570">A number theoretic series of I. Kasara</a>, Pacific J. Math., Volume 111, Number 2 (1984), 283-285.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 5.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">Euler-Mascheroni constant</a>

%F lim_{m -> infinity} ( Sum_{n=2..m} 1/(n*log(n)) -  log(log(m)/log(2)) ).

%e 0.4281657248712350751914588...

%t digits = 99; m0 = 10^digits; dm = 10^digits; Clear[g]; g[m_] := g[m] = NSum[1/(n*Log[n]) - (2*n*Log[Log[m]/Log[2]])/(-2 + m + m^2), {n, 2, m}, WorkingPrecision -> 2 digits, NSumTerms -> 1000, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 30}}]; g[m = m0]; g[m = m0 + dm]; While[Print["m = ", m // N // ScientificForm, " ", RealDigits[g[m], 10, digits]]; RealDigits[g[m], 10, digits + 2] != RealDigits[g[m - dm], 10, digits + 2], m = m + dm]; RealDigits[g[m], 10, digits] // First (* updated Apr 19 2016 *)

%Y Cf. A001620, A082633, A242616.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Aug 07 2014

%E Extended to 99 digits by _Jean-François Alcover_, Apr 19 2016