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A228725
Decimal expansion of the generalized Euler constant gamma(1,2).
19
6, 3, 5, 1, 8, 1, 4, 2, 2, 7, 3, 0, 7, 3, 9, 0, 8, 5, 0, 1, 1, 8, 7, 2, 1, 0, 5, 7, 7, 0, 2, 8, 9, 4, 9, 9, 5, 5, 8, 8, 2, 9, 7, 3, 5, 1, 5, 0, 0, 8, 9, 4, 2, 6, 4, 6, 3, 2, 2, 3, 6, 2, 2, 1, 8, 9, 1, 3, 0, 6, 7, 4, 3, 7, 3, 6, 7, 9, 6, 9, 3, 2, 7, 1
OFFSET
0,1
COMMENTS
The complement (A239097) is gamma(0,2) = lim_{x -> oo} (Sum_{0<n<=x, n even} (1/n - log(x)/2) = (A001620 - A002162)/2 = -0.05796575... - R. J. Mathar, Sep 06 2013
LINKS
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
J. C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013. See Section 3.8.
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), 125-142.
FORMULA
Equals lim_{x -> oo} (Sum_{0<n<=x, n odd} 1/n - log(x)/2).
Equals (A001620 + A002162)/2.
From Amiram Eldar, Jun 30 2020: (Start)
Equals -Integral_{x=0..1} log(log(1/x))*x dx.
Equals -Integral_{x=0..oo} exp(-2*x)*log(x) dx. (End)
Equals Integral_{x=0..1, y=0..1} log(-log(x*y))*x*y/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to one of Amiram Eldar's integrals.) - Petros Hadjicostas, Jun 30 2020
Equals -(psi(1/2) + log(2))/2 = (A020759 - A002162)/2. - Amiram Eldar, Jan 07 2024
EXAMPLE
0.63518142273073908501187210577028949955882973515008942646322...
MAPLE
(gamma+log(2))/2 ; evalf(%) ;
MATHEMATICA
RealDigits[(EulerGamma+Log[2])/2, 10, 120][[1]] (* Harvey P. Dale, Dec 26 2013 *)
PROG
(PARI) (Euler+log(2))/2 \\ Charles R Greathouse IV, Jul 21 2015
(Magma) SetDefaultRealField(RealField(100)); R:= RealField();
(EulerGamma + Log(2))/2; // G. C. Greubel, Aug 27 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
R. J. Mathar, Aug 31 2013
STATUS
approved