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A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.
(Formerly M3755 N1532)
805
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. - 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) * A082020 * A078585. - 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.

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.

B. Gugger, Problèmes corrigés de Mathématiques posés aux concours des Ecoles Militaires, Ecole de l'Air, 1992, option MP, 1ère épreuve, Ellipses, 1993, pp. 167-184.

J. Havil, Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.

J.-M. Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, Exercice 4.3.14, pages 371 and 387, 1997.

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).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1990.

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].

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/Pi^2 and into the formal enveloping series with rational coefficients only, arXiv:1501.00740 [math.NT], 2015-2016; Journal of Number Theory (Elsevier), Volume 158, pages 365-396, 2016.

D. Bradley, Ramanujan's formula for the logarithmic derivative of the Gamma function, arXiv:math/0505125 [math.CA], 2005.

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 [math.NA], Nov. 2013.

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

D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.

Chao-Ping Chen, Sharp inequalities and asymptotic series of a product related to the Euler-Mascheroni constant, Journal of Number Theory, Volume 165, August 2016, Pages 314-323.

E. Chlebus, A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant, Amer. Math. Mnthly, 118 (2011), 268-274.

M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, arXiv:1202.3093 [math.NT], 2012; Acta Appl. Math., 121 (2012), 1-3.

Dave's Math Tables, Gamma Constant

Philippe Deléham, Letter to N. J. A. Sloane, Apr 14 1997

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

Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016.

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

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

Kalpok Guha and Sourangshu Ghosh, Measuring Abundance with Abundancy Index, (2021).

Brady Haran and Tony Padilla, The mystery of 0.577, Numberphile video, 2016.

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

D. E. Knuth, Euler's constant to 1271 places, Math. Comp. 16 1962 275-281.

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, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.

M. Lerch, Expressions nouvelles de la constante d'Euler, S.-B. Kgl. Bohmischen Ges. Wiss., Article XLII (1897), Prague (5 pages).

Eric Naslund, Euler-Mascheroni constant expression, further simplification.

T. Papanikolaou, Plouffe's Inverter, Euler's constant to 1000000 decimals

S. Plouffe, using data from J. Borwein, 170000 digits of Euler or gamma constant [archived copy of a page on WorldWideSchool.org which doesn't exist any mode, cf. "Euler" link in left column].

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, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.

J. Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma, arXiv:math/0306008 [math.CA], 2003.

J. Sondow, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin, arXiv:math/0211075 [math.NT], 2002-2009; Math. Slovaca 59 (2009), 1-8.

J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; 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, arXiv:math/0610499 [math.CA], 2006; 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, arXiv:0705.0732 [math.NT], 2007; 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, arXiv:math/0304021 [math.NT], 2003; Ramanujan J. 12 (2006), 225-244.

D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.

Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant

Wikipedia, Stieltjes constants

A. Y. Yee, Large computations

Index entries for sequences related to Beatty sequences

FORMULA

Limit_{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.

Integral_{x=0..1} -log(log(1/x)). - Robert G. Wilson v, Jan 04 2006

Integral_{x=0..1,y=0..1} (x-1)/((1-x*y)*log(x*y)). - (see Sondow 2005)

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

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

Equals the lim_{n->infinity} fractional part of zeta(1+1/n). The corresponding fractional part for x->1 from below, using n-1/n, is -(1-a(n)). The fractional part found in this way for the first derivative of Zeta as x->1 is A252898. - Richard R. Forberg, Dec 24 2014

Limit_{x->1} (Zeta(x)-1/(x-1)) from Whittaker and Watson. 1990. - Richard R. Forberg, Dec 30 2014

exp(gamma) = lim_{i->infinity} exp(H(i)) - exp(H(i-1)), where H(i) = i-th Harmonic number. For a given n this converges faster than the standard definition, and two above, after taking the logarithm (e.g., 13 digits vs. 6 digits at n=3000000 or x=1+1/3000000). - Richard R. Forberg, Jan 08 2015

Limit_{n->infinity} (1/2) Sum_{j>=1} Sum_{k=1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n) (k + j*n))). - Dimitri Papadopoulos, Jan 13 2016

Equals 25/27 minus lim_{x->infinity} 2^(x+1)/3 - (22/27)*(4/3)^x - Zeta(Sum_{i>=1} (H_i/i^x)), letting H_i denote the i-th harmonic number. - John M. Campbell, Jan 29 2016

Limit_{x->0} -B'(x), where B(x) = -x zeta(1-x) is the "Bernoulli function". - Jean-François Alcover, May 20 2016

Sum_{k>=0} (1/2)(digamma(1/2+2^k) - digamma(2^k)) where digamma(x) = d/dx log(Gamma(x)). - Dimitri Papadopoulos, Nov 14 2016

a(n) = -10*floor(gamma*10^n) + floor(gamma*10^(n + 1)) for n >= 0. - Mariusz Iwaniuk, Apr 28 2017

Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma = -Pi*Integral_{0..infinity} a/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..infinity} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k) *a^(n-2*k). - Peter Luschny, Apr 19 2018

Limit_{s->0} (Zeta'(1-s)*s - Zeta(1-s)) / (Zeta(1-s)*s). - Peter Luschny, Jun 18 2018

log(2) * (gamma - (1/2) * log(2)) = -Sum_{v >= 1} (1/2^(v+1)) * (Delta^v (log(w)/w))|_{w=1}, where Delta(f(w)) = f(w) - f(w + 1) (forward difference). [This is a formula from Lerch (1897).] - Petros Hadjicostas, Jul 21 2019

From Amiram Eldar, Jul 05 2020: (Start)

Equals Integral_{x=1..oo} (1/floor(x) - 1/x) dx.

Equals Integral_{x=0..1} (1/(1-x) + 1/log(x)) dx = Integral_{x=0..1} (1/x + 1/log(1-x)) dx.

Equals -Integral_{-oo..oo} x*exp(x-exp(x)) dx.

Equals Sum_{k>=1} (-1)^k * floor(log_2(k))/k.

Equals (-1/2) * Sum_{k>=1} (Lambda(k)-1)/k, where Lambda is the Mangoldt function. (End)

Equals Integral_{0..1} -1/LambertW(-1,-x*exp(-x)) dx = 1 + Integral_{0..1} LambertW(-1/x*exp(-1/x)) dx. - Gleb Koloskov, Jun 12 2021

Equals Sum_{k>=2} (-1)^k * zeta(k)/k. - Vaclav Kotesovec, Jun 19 2021

Equals lim_{x->oo} log(x) - Sum_{p prime <= x} log(p)/(p-1). - Amiram Eldar, Jun 29 2021

Limit_{n->infinity} (2*HarmonicNumber(n) - HarmonicNumber(n^2)). After answer by Eric Naslund on Mathematics Stack Exchange, on Jun 21 2011. - Mats Granvik, Jul 19 2021

Equals Integral_{x=0..oo} ( exp(-x) * (1/(1-exp(-x)) - 1/x) ) dx (see Gugger or Monier). - Bernard Schott, Nov 21 2021

EXAMPLE

0.577215664901532860606512090082402431042...

MAPLE

Digits := 100; evalf(gamma);

MATHEMATICA

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

(1/2) N[Sum[PolyGamma[0, 1/2 + 2^k] - PolyGamma[0, 2^k], {k, 0, Infinity }], 30] (* Dimitri Papadopoulos, Nov 30 2016 *)

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

(Magma) EulerGamma(250); // G. C. Greubel, Aug 21 2018

(Python)

from sympy import S

def aupton(digs): return [int(d) for d in str(S.EulerGamma.n(digs+2))[2:-2]]

print(aupton(99)) # Michael S. Branicky, Nov 22 2021

CROSSREFS

Cf. A002852 (continued fraction).

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

Cf. A231095 (power tower using this constant).

Cf. A199332, A252898.

Denote the generalized Euler constants, also called Stieltjes constants, by Sti(n).

Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),

Sti(1/2) = A301816, Sti(1) = A082633 (cf. A262382/A262383), Sti(3/2) = A301817,

Sti(2) = A086279 (cf. A262384/A262385), Sti(3) = A086280 (cf. A262386/A262387),

Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,

Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

Sequence in context: A173930 A154802 A210624 * A242220 A245726 A101456

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 September 28 01:47 EDT 2022. Contains 357063 sequences. (Running on oeis4.)