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A001620
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Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.
(Formerly M3755 N1532)
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919
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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
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OFFSET
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0,1
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COMMENTS
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Yee (2010) computed 29844489545 decimal digits of gamma.
Decimal expansion of 0th Stieltjes constant. - Paul Muljadi, Aug 24 2010
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REFERENCES
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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.
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LINKS
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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].
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FORMULA
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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,y=0..1} (x-1)/((1-x*y)*log(x*y)). - (see Sondow 2005)
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
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 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
Equals 1/2 + Limit_{s->1} (Zeta(s) + Zeta(1/s))/2. - Thomas Ordowski, Jan 12 2023
Equals Sum_{j>=2} Sum_{k>=2} ((k-1)/(k*j^k)). - Mike Tryczak, Apr 06 2023
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EXAMPLE
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0.577215664901532860606512090082402431042...
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MAPLE
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Digits := 100; evalf(gamma);
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MATHEMATICA
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(1/2) N[Sum[PolyGamma[0, 1/2 + 2^k] - PolyGamma[0, 2^k], {k, 0, Infinity }], 30] (* Dimitri Papadopoulos, Nov 30 2016 *)
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PROG
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(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
(Python)
from sympy import S
def aupton(digs): return [int(d) for d in str(S.EulerGamma.n(digs+2))[2:-2]]
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CROSSREFS
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Cf. A231095 (power tower using this constant).
Denote the generalized Euler constants, also called Stieltjes constants, by Sti(n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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