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A073003
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Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant.
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21
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5, 9, 6, 3, 4, 7, 3, 6, 2, 3, 2, 3, 1, 9, 4, 0, 7, 4, 3, 4, 1, 0, 7, 8, 4, 9, 9, 3, 6, 9, 2, 7, 9, 3, 7, 6, 0, 7, 4, 1, 7, 7, 8, 6, 0, 1, 5, 2, 5, 4, 8, 7, 8, 1, 5, 7, 3, 4, 8, 4, 9, 1, 0, 4, 8, 2, 3, 2, 7, 2, 1, 9, 1, 1, 4, 8, 7, 4, 4, 1, 7, 4, 7, 0, 4, 3, 0, 4, 9, 7, 0, 9, 3, 6, 1, 2, 7, 6, 0, 3, 4, 4, 2, 3, 7
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OFFSET
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0,1
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COMMENTS
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0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) Sum_{n>=0} (-y)^n n! = KummerU(1,1,1/y)/y.
Decimal expansion of phi(1) where phi(x) = Integral_{t>=0} e^-t/(x+t) dt. - Benoit Cloitre, Apr 11 2003
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m => -1, is intimately related to Gompertz's constant. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A000110 the Bell numbers and A040027 a sequence that was published by Gould, see for more information A163940. - Johannes W. Meijer, Oct 16 2009
Named by Le Lionnais (1983) after the English self-educated mathematician and actuary Benjamin Gompertz (1779 - 1865). It was named the Euler-Gompertz constant by Finch (2003). Lagarias (2013) noted that he has not located this constant in Gompertz's writings. - Amiram Eldar, Aug 15 2020
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REFERENCES
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Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171
Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 424-425.
Francois Le Lionnais, Les nombres remarquables, Paris: Hermann, 1983. See p. 29.
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948, p. 356.
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LINKS
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Robert Price, Table of n, a(n) for n = 0..10000
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009. [From R. J. Mathar, Feb 14 2009]
Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
G. H. Hardy, Divergent Series , Oxford University Press, 1949. p. 29. - Johannes W. Meijer, Oct 16 2009
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628, preprint, arXiv:1303.1856 [math.NT], 2013.
István Mezo, Gompertz constant, Gregory coefficients and a series of the logarithm function, Journal of Analysis & Number Theory, Vol. 2, No. 2 (2014), pp. 33-36.
Simon Plouffe, -exp(1)*Ei(-1)
Tanguy Rivoal, On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant, Michigan Math. J., Vol. 61, No. 2 (2012), pp. 239-254.
Ed Sandifer, Divergent Series, How Euler Did It, MAA Online, June 2006. - Johannes W. Meijer, Oct 16 2009
Eric Weisstein's World of Mathematics, Gompertz Constant
Eric Weisstein's World of Mathematics, Exponential Integral
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FORMULA
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phi(1) = e*(Sum_{k>=1} (-1)^(k-1)/(k*k!) - Gamma) = 0.596347362323194... where Gamma is the Euler constant.
G = 0.596347... = 1/(1+1/(1+1/(1+2/(1+2/(1+3/(1+3/(1+4/(1+4/(1+5/(1+5/(1+6/(... - Philippe Deléham, Aug 14 2005
Equals A001113*A099285. - Johannes W. Meijer, Oct 16 2009
From Peter Bala, Oct 11 2012: (Start)
Stieltjes found the continued fraction representation G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793 and the denominators in A002720.
Also, 1 - G has the continued fraction representation 1/(3 - 2/(5 - 6/(7 - ... -n*(n+1)/((2*n+3) - ...)))) with convergents beginning [1/3, 5/13, 29/73, 201/501, ...]. The numerators are in A201203 (unsigned) and the denominators are in A000262.
(End)
G = f(1) with f solution to the o.d.e. x^2*f'(x) + (x+1)*f(x)=1 such that f(0)=1. - Jean-François Alcover, May 28 2013
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..1} 1/(1-log(x)) dx.
Equals Integral_{x=1..oo} exp(1-x)/x dx.
Equals Integral_{x=0..oo} exp(-x)*log(x+1) dx.
Equals Integral_{x=0..oo} exp(-x)/(x+1) dx. (End)
From Gleb Koloskov, May 01 2021: (Start)
Equals Integral_{x=0..1} LambertW(e/x)-1 dx.
Equals Integral_{x=0..1} 1+1/LambertW(-1,-x/e) dx. (End)
Equals lim_{n->infinity} A040027(n)/A000110(n+1). - Vaclav Kotesovec, Feb 22 2021
G = lim_{n -> infinity} A321942(n)/A000262(n). - Peter Bala, Mar 21 2022
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EXAMPLE
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0.59634736232319407434107849936927937607417786015254878157348491...
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MATHEMATICA
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RealDigits[N[-Exp[1]*ExpIntegralEi[-1], 105]][[1]]
G = 1/Fold[Function[2*#2 - #2^2/#1], 2, Reverse[Range[10^4]]] // N[#, 105]&; RealDigits[G] // First (* Jean-François Alcover, Sep 19 2014 *)
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PROG
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(PARI) eint1(1)*exp(1) \\ Charles R Greathouse IV, Apr 23 2013
(MAGMA) SetDefaultRealField(RealField(100)); ExponentialIntegralE1(1)*Exp(1); // G. C. Greubel, Dec 04 2018
(Sage) numerical_approx(exp_integral_e(1, 1)*exp(1), digits=100) # G. C. Greubel, Dec 04 2018
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CROSSREFS
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Cf. A000522 (arrangements), A001620, A000262, A002720, A002793, A058006 (alternating factorial sums), A153229, A201203, A283743 (Ei(1)/e), A321942.
Sequence in context: A051158 A117605 A303983 * A344093 A087498 A274633
Adjacent sequences: A073000 A073001 A073002 * A073004 A073005 A073006
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KEYWORD
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cons,nonn
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AUTHOR
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Robert G. Wilson v, Aug 03 2002
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EXTENSIONS
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Additional references from Gerald McGarvey, Oct 10 2005
Link corrected by Johannes W. Meijer, Aug 01 2009
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STATUS
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approved
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