A228725 Decimal expansion of the generalized Euler constant gamma(1,2).

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

G. C. Greubel, Table of n, a(n) for n = 0..10000

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

Cf. A001620, A002162, A020759, A239097.

Sequence in context: A195471 A305187 A065418 * A286982 A153595 A195482

Adjacent sequences: A228722 A228723 A228724 * A228726 A228727 A228728

Keyword

nonn,cons

Author

R. J. Mathar, Aug 31 2013

Status

approved