A350763 Decimal expansion of gamma + log(2), where gamma is Euler's constant (A001620).

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, 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, 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

Offset: 1

Author

Amiram Eldar, Jan 14 2022

Keywords

• nonn
• cons

Examples

			1.2703628454614781700237442115405789991176594703...
		

References

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

Links

• Philippe Flajolet and Ilan Vardi, Zeta function expansions of classical constants, 1996.
• Xavier Gourdon and Pascal Sebah, Collection of formulae for Euler's constant gamma, 2008.
• Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 18, Table 1.
• Wikipedia, Harmonic numbers for real and complex values.

Crossrefs

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

Programs

Mathematica

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

Formula

Equals A001620 + A002162.

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

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

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

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

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

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.

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

Equals 2*A228725. - Hugo Pfoertner, Jul 03 2024