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A269330
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Decimal expansion of the "alternating Euler constant" beta = li(2) - gamma.
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3
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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!).
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EXAMPLE
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0.4679481152159599242380767991122107054804562422112779...
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MAPLE
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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
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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]]
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PROG
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(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)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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