Search: a069284 -id:a069284
|
| |
|
|
A269330
|
|
Decimal expansion of the "alternating Euler constant" beta = li(2) - gamma.
|
|
+10
3
|
|
|
|
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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,1
|
|
|
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.
|
|
|
LINKS
|
|
|
|
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!).
|
|
|
EXAMPLE
|
0.4679481152159599242380767991122107054804562422112779...
|
|
|
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]]
|
|
|
PROG
|
(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)))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A069285
|
|
Continued fraction for Li(2).
|
|
+10
2
|
|
|
|
1, 22, 7, 16, 1, 1, 2, 1, 4, 6, 1, 40, 1, 1, 1, 1, 1, 1, 3, 9, 1, 1, 1, 1, 35, 7, 9, 2, 1, 3, 8, 2, 5, 1, 1, 4, 1, 1, 6, 1, 18, 14, 1, 10, 1, 3, 1, 1, 5, 4, 1, 59, 1, 7, 2, 1, 1, 7, 1, 43, 12, 21, 2, 4, 8, 16, 1, 22, 4, 37, 1, 1, 1, 12, 1, 1, 1, 3, 3, 1, 1, 1, 4, 1, 3, 5, 3, 1, 33, 2, 15, 5, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
1.0451637801174927848445888891946131365226155781512015758329...
|
|
|
MATHEMATICA
|
ContinuedFraction[LogIntegral[2], 100] (* Harvey P. Dale, Dec 31 2015 *)
|
|
|
PROG
|
(PARI) contfrac(-real(eint1(-log(2)))) \\ G. C. Greubel, Apr 22 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,cofr
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.004 seconds
|
