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A001620 Decimal expansion of Euler's constant (or Euler-Mascheroni constant) gamma.
(Formerly M3755 N1532)
242
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; text; internal format)
OFFSET

0,1

COMMENTS

Yee (2010) computed 29844489545 decimal digits of gamma.

Decimal expansion of 0th Stieltjes constant. [From Paul Muljadi, Aug 24 2010]

The value of Euler's constant is close to (18/Pi^2)*sum(n>=0, 1/4^(2^n)) = 0.5770836328... = 6/5 * constant in A082020 * constant in A078585. [From Arkadiusz Wesolowski, Mar 27 2012]

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.

D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. [From N. J. A. Sloane, Mar 24 2012]

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.

S. Sykora, Blazys Expansions and Continued Fractions, Stan's Library, Volume IV, Mathematics, 2013; http://www.ebyte.it/stan/2013_BlazysExpansions.pdf

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

R. P. Brent and F. Johansson,A bound for the error term in the Brent-McMillan algorithm, arXiv 1312.0039, Nov. 2013.

C. K. Caldwell, The Prime Glossary, Euler's constant

M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., 121 (2012), 1-3.

Dave's Math Tables, Gamma Constant

Thomas and Joseph Dence, A survey of Euler's constant, Math. Mag., 82 (2009), 255-265.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants

X. Gourdon and P. Sebah, The Euler's constant gamma

J. C. Kluyver, Euler's constant and natural numbers, Proc. K. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

Stefan Krämer, Euler's Constant γ=0.577... Its Mathematics and History

Richard Kreckel, 116 million digits of Euler's constant (bzipped)

A. Krowne, PlanetMath.org, Euler's constant

Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628.

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 S. Zlobin, Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant, Math. Notes, 84 (2008), 568-583, Erratum p. 887.

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, Aug 24 2010]

A. Y. Yee, Large computations

Index entries for sequences related to Beatty sequences

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)). - [ 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).

Integrate_{x=0..infinity} -log(x)*exp(-x). [Jean-François Alcover, Mar 22 2013]

Integrate_{x=0..1} (1-exp(-x)-exp(-1/x))/x. [Jean-François Alcover, Apr 11 2013]

EXAMPLE

0.577215664901532860606512090082402431042...

MAPLE

Digits := 100; evalf(gamma);

MATHEMATICA

RealDigits[ EulerGamma, 10, 105][[1]] (* Robert G. Wilson v, Nov 01 2004 *)

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)); } /* Harry J. Smith, Apr 15 2009 */

CROSSREFS

Cf. A002852 (continued fraction).

Cf. A073004 (exp(gamma)) and A094640 ("alternating Euler constant").

Cf. A199332.

Cf. A231095 (power tower using this constant).

Sequence in context: A173930 A154802 A210624 * A101456 A084823 A117034

Adjacent sequences:  A001617 A001618 A001619 * A001621 A001622 A001623

KEYWORD

nonn,cons,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 18 03:34 EDT 2014. Contains 240688 sequences.