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A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant. 22
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
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.
LINKS
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009. - 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.
Michael Penn, why some series are "regularizable", YouTube video (2023).
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
FORMULA
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
Equals Sum_{n >= 1} 1/(n*L(n, -1)*L(n-1, -1)), where L(n, x) denotes the n-th Laguerre polynomial. This is the case x = 1 of the identity Integral_{t >= 0} exp(-t)/(x + t) dt = Sum_{n >= 1} 1/(n*L(n, -x)*L(n-1, -x)) valid for Re(x) > 0. - Peter Bala, Mar 21 2024
EXAMPLE
0.59634736232319407434107849936927937607417786015254878157348491...
MATHEMATICA
RealDigits[N[-Exp[1]*ExpIntegralEi[-1], 105]][[1]]
(* Second program: *)
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 *)
PROG
(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
CROSSREFS
Cf. A000522 (arrangements), A001620, A000262, A002720, A002793, A058006 (alternating factorial sums), A153229, A201203, A283743 (Ei(1)/e), A321942, A369883.
Sequence in context: A051158 A117605 A303983 * A344093 A087498 A274633
KEYWORD
cons,nonn,changed
AUTHOR
Robert G. Wilson v, Aug 03 2002
EXTENSIONS
Additional references from Gerald McGarvey, Oct 10 2005
Link corrected by Johannes W. Meijer, Aug 01 2009
STATUS
approved

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Last modified April 16 01:01 EDT 2024. Contains 371696 sequences. (Running on oeis4.)