A269330 Decimal expansion of the "alternating Euler constant" beta = li(2) - gamma.

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

Offset: 0

Author

Iaroslav V. Blagouchine, Feb 23 2016

Keywords

• nonn
• cons

Comments

The function li(x) is the integral logarithm, gamma is Euler's constant.

Decimal expansion of Sum_{n>=1} G_n/n = beta, where numbers G_n are Gregory's coefficients (see A002206 and A002207). In comparison to the Fontana-Mascheroni's series Sum_{n>=1} |G_n|/n = gamma (see A195189), the constant beta may be regarded as the "alternating Euler constant". A similar analogy also exists between gamma and log(4/Pi), see A094640.

Another striking analogy between beta and gamma follows from the fact that beta = Integral_{x=0..1} (1/log(1+x) - 1/x) dx, while gamma = Integral_{x=0..1} (1/log(1-x) + 1/x) dx.

For more details, see references below.

Examples

			0.4679481152159599242380767991122107054804562422112779...
		

Links

• Iaroslav V. Blagouchine, Table of n, a(n) for n = 0..1000
• Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
• Iaroslav V. Blagouchine, Another Alternating Analogue of Euler's Constant. The American Mathematical Monthly, vol. 120, no. 1, pp. 24-34, 2022.
• Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578v3 [math.HO], 2022.

Crossrefs

Cf. A001620, A002206, A002207, A069284, A094640, A195189, A270857, A270859.

Programs

Maple

evalf(Li(2)-gamma, 120)
evalf(Ei(ln(2))-gamma, 120)
evalf(int(1/ln(1+x)-1/x, x = 0..1), 120)
evalf(ln(ln(2))+sum(ln(2)^k/(k*factorial(k)), k = 1..infinity), 120)

Mathematica

RealDigits[LogIntegral[2] - EulerGamma, 10, 120][[1]]
RealDigits[ExpIntegralEi[Log[2]] - EulerGamma, 10, 120][[1]]
RealDigits[Integrate[1/Log[1+x] - 1/x, {x, 0, 1}], 10, 120][[1]]
RealDigits[Log[Log[2]] + Sum[Log[2]^k/(k*k!), {k, 1, ∞}], 10, 120][[1]]

PARI

default(realprecision, 120); -real(eint1(-log(2)))-Euler

PARI

default(realprecision, 120); intnum(x=0,1,1/log(1+x)-1/x) \\ Note: PARI/GP v. 2.7.3 is able to compute only 19 digits

PARI

default(realprecision,120); log(log(2))+sumpos(k=1,log(2)^k/(k*factorial(k)))

Formula

Equals li(2) - gamma.

Equals Ei(log(2)) - gamma.

Equals Integral_{x=0..1} (1/log(1+x) - 1/x) dx.

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