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
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.
Programs
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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)
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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
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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
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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!).
Comments