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A001620
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Decimal expansion of Euler's constant (or Euler-Mascheroni constant) gamma.
(Formerly M3755 N1532)
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157
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5, 7, 7, 2, 1, 5, 6, 6, 4, 9, 0, 1, 5, 3, 2, 8, 6, 0, 6, 0, 6, 5, 1, 2, 0, 9, 0, 0, 8, 2, 4, 0, 2, 4, 3, 1, 0, 4, 2, 1, 5, 9, 3, 3, 5, 9, 3, 9, 9, 2, 3, 5, 9, 8, 8, 0, 5, 7, 6, 7, 2, 3, 4, 8, 8, 4, 8, 6, 7, 7, 2, 6, 7, 7, 7, 6, 6, 4, 6, 7, 0, 9, 3, 6, 9, 4, 7, 0, 6, 3, 2, 9, 1, 7, 4, 6, 7, 4, 9
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Yee (2010) computed 29844489545 decimal digits of gamma.
Decimal expansion of 0th Stieltjes constant. [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Aug 24 2010]
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 3.
E. Chlebus, A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant, Amer. Math. Mnthly, 118 (2011), 268-274.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 28-40.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.
J. Havil, Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.
D. E. Knuth, Euler's constant to 1271 places. Math. Comp. 16 1962 275-281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Borwein, 170000 digits of Euler or gamma constant
D. Bradley, Ramanujan's formula for the logarithmic derivative of the Gamma function
R. P. Brent, Ramanujan and Euler's constant
C. K. Caldwell, The Prime Glossary, Euler's constant
Dave's Math Tables, Gamma Constant
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
X. Gourdon and P. Sebah, The Euler's constant gamma
Richard Kreckel, 116 million digits of Euler's constant (bzipped)
A. Krowne, PlanetMath.org, Euler's constant
T. Papanikolaou, Plouffe's Inverter, Euler's constant to 1000000 decimals
S. Ramanujan, A series for Euler's constant, Messenger of Math., 46 (1917), 73-80.
S. Ramanujan, Question 327, J. Ind. Math. Soc.
J. Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
J. Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
J. Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
J. Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma
J. Sondow, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin, Math. Slovaca 59 (2009), 1-8.
J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (1) (2007), 292-314.
J. Sondow and W. Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006), 225-244.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant
Wikipedia, Stieltjes constants [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Aug 24 2010]
A. Y. Yee, Large computations
Index entries for sequences related to Beatty sequences
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FORMULA
| Lim_{n->infinity} (1 + 1/2 + ... + 1/n - log(n)) (definition).
sum(n>=1, (1/n - log(1 + 1/n)) ), since log(1 + 1/1) + ... + log(1 + 1/n) telescopes to log(n+1) and Lim_{n->infinity} (log(n+1) - log(n)) = 0.
Integrate_{x=0..1} -log(log(1/x)). - (from Robert G. Wilson v Jan 04 2006)
Integrate_{x=0..1,y=0..1} (x-1)/((1-x*y)*log(x*y)) - (see Sondow 2005).
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EXAMPLE
| .577215664901532860606512090082402431042...
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MAPLE
| Digits := 100; evalf(gamma);
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MATHEMATICA
| RealDigits[ EulerGamma, 10, 105][[1]] (from Robert G. Wilson v Nov 01 2004)
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PROG
| (PARI) { default(realprecision, 20080); x=Euler; d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b001620.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 15 2009]
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CROSSREFS
| Cf. A002852 (continued fraction).
See also A073004 (exp(gamma)) and A094640 ("alternating Euler constant").
Cf. A199332.
Sequence in context: A197257 A173930 A154802 * A101456 A084823 A117034
Adjacent sequences: A001617 A001618 A001619 * A001621 A001622 A001623
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KEYWORD
| nonn,cons,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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