A350763 - OEIS

%I #15 Jul 03 2024 10:05:19

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

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

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

%N Decimal expansion of gamma + log(2), where gamma is Euler's constant (A001620).

%D J. C. Kluyver, De constante van Euler en de natuurlijke getallen, Amst. Ak. Versl., Vol. 33 (1924), pp. 149-151.

%H Philippe Flajolet and Ilan Vardi, <a href="http://algo.inria.fr/flajolet/Publications/FlVa96.pdf">Zeta function expansions of classical constants</a>, 1996.

%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.pdf">Collection of formulae for Euler's constant gamma</a>, 2008.

%H Alessandro Languasco and Pieter Moree, <a href="https://arxiv.org/abs/2406.16547">Euler constants from primes in arithmetic progression</a>, arXiv:2406.16547 [math.NT], 2024. See p. 18, Table 1.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_number#Harmonic_numbers_for_real_and_complex_values">Harmonic numbers for real and complex values</a>.

%F Equals A001620 + A002162.

%F Equals 1 + Sum_{k>=2} ((-1)^k * (zeta(k)-1)/k).

%F Equals 3/2 - Sum_{k>=2} ((-1)^k * (k-1) * (zeta(k)-1)/k) (Flajolet and Vardi, 1996).

%F Equals 5/4 - (1/2) * Sum_{k>=3} ((-1)^k * (k-1) * (zeta(k)-1)/k) (Gourdon and Sebah, 2008).

%F Equals 1 + Sum_{k>=2} (1/k - log(1+1/k)).

%F Equals 1 + Sum_{k>=0} abs(A002206(k))/((k+1)*(k+2)*A002207(k)) (Kluyver, 1924).

%F Equal Integral_{x>=0} (1/(1+x^2/4) - cos(x))/x dx = Integral_{x>=0} (1/(1+x^2) - cos(2*x))/x dx.

%F Equals Integral_{x=1..2} H(x) dx, where H(x) is the harmonic number for real variable x.

%F Equals 2*A228725. - _Hugo Pfoertner_, Jul 03 2024

%e 1.2703628454614781700237442115405789991176594703...

%t RealDigits[EulerGamma + Log[2], 10, 100][[1]]

%Y Cf. A001620, A002162, A002206, A002207, A228725.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Jan 14 2022