cp's OEIS Frontend

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

%I A228725 #64 Feb 24 2025 02:04:12
%S A228725 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,
%T A228725 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,
%U A228725 0,6,7,4,3,7,3,6,7,9,6,9,3,2,7,1
%N A228725 Decimal expansion of the generalized Euler constant gamma(1,2).
%C A228725 The complement (A239097) is gamma(0,2) = lim_{x->oo} ((Sum_{n=1..x, n even} 1/n) - log(x)/2) = (A001620 - A002162)/2 = -0.05796575... - _R. J. Mathar_, Sep 06 2013
%H A228725 G. C. Greubel, <a href="/A228725/b228725.txt">Table of n, a(n) for n = 0..10000</a>
%H A228725 M. L. Glasser, <a href="https://doi.org/10.1080/00029890.2019.1565856">A note on Beukers's and related integrals</a>, Amer. Math. Monthly 126(4) (2019), 361-363.
%H A228725 J. C. Lagarias, <a href="http://arxiv.org/abs/1303.1856">Euler's constant: Euler's work and modern developments</a>, arXiv:1303.1856 [math.NT], 2013. See Section 3.8.
%H A228725 D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf">Euler constants for arithmetic progressions</a>, Acta Arith. 27 (1975), 125-142.
%F A228725 Equals lim_{x->oo} ((Sum_{n=1..x, n odd} 1/n) - log(x)/2).
%F A228725 Equals (A001620 + A002162)/2.
%F A228725 From _Amiram Eldar_, Jun 30 2020: (Start)
%F A228725 Equals -Integral_{x=0..1} log(log(1/x))*x dx.
%F A228725 Equals -Integral_{x=0..oo} exp(-2*x)*log(x) dx. (End)
%F A228725 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
%F A228725 Equals -(psi(1/2) + log(2))/2 = (A020759 - A002162)/2. - _Amiram Eldar_, Jan 07 2024
%e A228725 0.63518142273073908501187210577028949955882973515008942646322...
%p A228725 (gamma+log(2))/2 ; evalf(%) ;
%t A228725 RealDigits[(EulerGamma+Log[2])/2,10,120][[1]] (* _Harvey P. Dale_, Dec 26 2013 *)
%o A228725 (PARI) (Euler+log(2))/2 \\ _Charles R Greathouse IV_, Jul 21 2015
%o A228725 (Magma) SetDefaultRealField(RealField(100)); R:= RealField();
%o A228725 (EulerGamma + Log(2))/2; // _G. C. Greubel_, Aug 27 2018
%Y A228725 Cf. A001620, A002162, A020759, A239097.
%K A228725 nonn,cons
%O A228725 0,1
%A A228725 _R. J. Mathar_, Aug 31 2013