A241005 Decimal expansion of gamma', the analog of Euler's constant when 1/x is replaced by 1/(x*log(x)).

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Aug 07 2014

Keywords

• nonn
• cons

Examples

			0.4281657248712350751914588...
		

References

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

Links

• Harold G. Diamond, A number theoretic series of I. Kasara, Pacific J. Math., Volume 111, Number 2 (1984), 283-285.
• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 5.
• Eric Weisstein's MathWorld, Euler-Mascheroni Constant
• Wikipedia, Euler-Mascheroni constant

Crossrefs

Cf. A001620, A082633, A242616.

Programs

Mathematica

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 *)

Formula

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

Extensions

Extended to 99 digits by Jean-François Alcover, Apr 19 2016