A058655 -id:A058655 - cp's OEIS frontend

A229191 Decimal expansion of the integral_{x=0..Infinity} 1/x^x dx.

1, 9, 9, 5, 4, 5, 5, 9, 5, 7, 5, 0, 0, 1, 3, 8, 0, 0, 0, 4, 1, 8, 7, 2, 4, 6, 9, 8, 4, 5, 2, 7, 2, 4, 3, 5, 2, 0, 8, 6, 2, 1, 6, 6, 3, 6, 9, 6, 7, 9, 7, 8, 8, 7, 2, 7, 8, 8, 3, 0, 0, 0, 6, 0, 9, 8, 3, 0, 3, 1, 6, 1, 7, 1, 4, 6, 5, 6, 6, 3, 6, 3, 0, 6, 6, 9, 5, 4, 9, 2, 7, 7, 8, 9, 4, 6, 3, 8, 7, 7, 0, 5, 8, 1, 6, 7, 6, 3, 7, 7, 0

Offset: 1

Robert G. Wilson v, Sep 15 2013

"The function x^x grows even more quickly than Gamma(x) and the integral {0-inf} 1/x^x dx = 1.9954559575... and the integral {1-inf} 1/x^x dx = 0.7041699604... ." [Finch]

			1.9954559575001380004187246984527243520862166369679788727883...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 263.

Cf. A058655, A073009, A245637.

evalf(int(1/x^x, x=0..infinity), 120);  # Alois P. Heinz, Dec 30 2021

RealDigits[ NIntegrate[ 1/x^x, {x, 0, 100}, MaxRecursion -> 5000, MaxPoints -> 5000, AccuracyGoal-> 111, PrecisionGoal -> 111, WorkingPrecision -> 120], 10, 111][[1]]

A046943 Continued fraction for Fransen-Robinson constant Integral_{x>=0} 1/Gamma(x).

Original entry on oeis.org

2, 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 10, 1, 4, 7, 2, 2, 2, 46, 18, 1, 1, 3, 1, 1, 4, 5, 1, 1, 28, 6, 2, 1, 23, 1, 6, 1, 18, 1, 4, 1, 2, 1, 3, 2, 3, 5, 1, 1, 7, 1, 1, 1, 8, 1, 1, 1, 1, 2, 1, 7, 2, 2, 1, 1, 1, 1, 6, 1, 2, 2, 11, 2, 1, 1, 3, 7, 1

Offset: 0

Wouter Meeussen

			2.807770242028519365221501186... = 2 + 1/(1 + 1/(4 + 1/(4 + 1/(1 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..1033
A. Fransen, Accurate Determination of the Inverse Gamma Integral, Nordisk Tidskr. Informationsbehandling (BIT) 19, 137-138, 1979.
A. Fransen and S. Wrigge, High-Precision Values of the Gamma Function and of Some Related Coefficients, Math. Comput. 34, 553-566, 1980.
A. Fransen and S. Wrigge, Addendum and Corrigendum to 'High-Precision Values of the Gamma Function and of Some Related Coefficients' Math. Comput. 37, 233-235, 1981.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A058655 (decimal expansion). - Harry J. Smith, May 13 2009

f := N[Integrate[1/Gamma[x], {x, 0, Infinity}], 55]; ContinuedFraction[f, 50] (* G. C. Greubel, Nov 06 2017 *)

A247377 Decimal expansion of Integral_{0..oo} 1/Gamma(1+x) dx, a variation of the Fransén-Robinson constant.

Original entry on oeis.org

2, 2, 6, 6, 5, 3, 4, 5, 0, 7, 6, 9, 9, 8, 4, 8, 8, 3, 5, 0, 7, 1, 9, 6, 3, 8, 5, 7, 6, 7, 8, 2, 2, 0, 9, 1, 8, 4, 0, 8, 8, 2, 9, 7, 1, 4, 2, 8, 0, 2, 2, 2, 1, 9, 3, 8, 6, 1, 0, 9, 3, 5, 5, 4, 4, 2, 9, 1, 8, 8, 9, 9, 7, 6, 9, 1, 3, 7, 5, 2, 8, 1, 0, 8, 5, 0, 9, 1, 0, 6, 9, 7, 4, 7, 9, 3, 4, 0, 6, 9, 5, 8, 8, 4

Offset: 1

Jean-François Alcover, Sep 15 2014

			2.266534507699848835071963857678220918408829714280222193861...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.6 Fransén-Robinson constant, p. 263.

Eric Weisstein's MathWorld, Fransén-Robinson Constant

Cf. A058655.

NIntegrate[1/Gamma[1 + x], {x, 0, Infinity}, WorkingPrecision -> 104] // RealDigits // First

localprec(100); intnum(x=0,[[1], 1],1/gamma(1+x)) \\ Dumitru Damian, Oct 12 2023

from mpmath import *
mp.dps = 200
A247377 = [d for d in nstr(quad(lambda x:1/gamma(1+x),[0,inf]),n=mp.dps)[:-1] if d != '.'] # Chai Wah Wu, Sep 16 2014

Also equals e - Integral_{-oo..oo} e^(-e^x)/(x^2 + Pi^2) dx (observed by Ramanujan).

A273017 Decimal expansion of the first moment of the reciprocal gamma distribution.

Original entry on oeis.org

1, 9, 3, 4, 5, 6, 7, 0, 4, 2, 1, 4, 7, 8, 8, 4, 7, 2, 1, 1, 8, 3, 7, 1, 4, 7, 0, 4, 3, 6, 9, 1, 7, 8, 9, 2, 4, 3, 8, 2, 1, 7, 5, 5, 9, 2, 2, 6, 6, 5, 8, 8, 4, 8, 3, 8, 5, 5, 4, 4, 7, 5, 4, 2, 2, 5, 9, 5, 4, 4, 0, 8, 7, 4, 7, 1, 0, 1, 8, 2, 4, 7, 2, 2, 5, 4, 4, 5, 0, 0, 3, 8, 3, 4, 8, 2, 1, 0, 1, 7

Offset: 1

Jean-François Alcover, May 13 2016

			1.93456704214788472118371470436917892438217559226658848385544754...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.6 Fransén-Robinson constant, p. 262.

Steven R. Finch Errata and Addenda to Mathematical Constants p. 35.
Eric Weisstein's World of Mathematics, Fransén-Robinson Constant
Wikipedia, Reciprocal gamma function

Cf. A058655.

digits = 100;
I0 = NIntegrate[1/Gamma[x], {x, 0, Infinity}, WorkingPrecision -> digits + 5];
M1 = (1/I0) NIntegrate[x/Gamma[x], {x, 0, Infinity}, WorkingPrecision -> digits + 5];
RealDigits[M1, 10, digits][[1]]

default(realprecision, 120); intnum(x=0, [[1], 1], x/gamma(x))/intnum(x=0, [[1], 1], 1/gamma(x)) \\ Vaclav Kotesovec, May 14 2016

(1/I)*Integral_{x>=0} x/gamma(x) dx where I = Integral_{x>=0} 1/gamma(x) dx is the Fransén-Robinson constant.

A360375 Decimal expansion of the area under the curve of the reciprocal of the Hadamard gamma function from zero to infinity.

Original entry on oeis.org

3, 3, 6, 8, 2, 0, 2, 9, 2, 9, 6, 0, 7, 0, 2, 2, 7, 9, 2, 1, 6, 2, 2, 0, 5, 9, 6, 2, 2, 0, 9, 3, 6, 2, 5, 4, 8, 4, 7, 6, 1, 0, 6, 4, 8, 8, 7, 6, 1, 0, 3, 1, 2, 1, 9, 4, 7, 0, 2, 8, 7, 5, 2, 0, 2, 6, 1, 6, 1, 6, 0, 5, 1, 3, 3, 6, 1, 3, 1, 4, 4, 2, 0, 3, 0, 2, 5, 3, 9, 3, 9, 8, 4, 1, 2, 4, 4, 3, 8, 1, 3, 8, 1, 7, 2

Offset: 1

Hywel Normington, Feb 04 2023

Close to 3 + (1/e) = 3.367879...

Sum_{n>=0} 1/H(n) = 1/log(2) + e - 1 = 3.1609768... This integral may have a similar representation to the Fransen-Robinson constant.

			3.368202929607022792162205962209362548476...

J. Hadamard, (1894), Oeuvre de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, 1968.

Philip J. Davis, Leonhard Euler's Integral: A Historical Profile Of The Gamma Function, Am. Math. Monthly 66, 849-869 (1959).
J. Hadamard, Sur l’Expression du Produit 1*2*3...*(n-1) par une Fonction Entière (in French, obtainable from Peter Luschny's Website).
Hywel Normington, Reciprocal Gamma Integrals, 2023.
Hywel Normington, Julia code, 2023.
Hywel Normington, Python code, 2023.
Wikipedia, Hadamard's gamma function.

Cf. A058655.

H := x -> 1/((sin(x*Pi)*(Psi(x/2) - Psi(1/2 + x/2)) + 2*Pi) * GAMMA(x)):
evalf[80](2*Pi*Int(H, 0..60, method = Gquad)); # _Peter Luschny, Feb 20 2023

RealDigits[NIntegrate[2*Gamma[1-x]/(PolyGamma[0, 1 - x/2] - PolyGamma[0, 1/2 - x/2]), {x, 0, Infinity}, WorkingPrecision -> 105, MaxRecursion -> Infinity]][[1]] (* Vaclav Kotesovec, Feb 19 2023 *)

default(realprecision, 200); intnum(x=0,[[1], 1], 2*gamma(1-x) / (psi(1-x/2) - psi(1/2-x/2))) \\ (default(realprecision, 200) is enough for 40 valid digits, \p 500 for 71 valid digits, \p 1000 for 110 valid digits, \p 2000 for 171 valid digits). - Vaclav Kotesovec, Feb 19 2023

Hadamard function definitions:
  H(x) = (1/Gamma(1-x)) * (d/dx) log(Gamma(1/2 - x/2)/Gamma(1-x/2)).
  H(x) = Gamma(x)*(1 + (sin(Pi*x)/(2*Pi)) * (Psi(x/2) - Psi((x+1)/2))).
Equals Integral_{0..oo} 1/H(x) dx.

More digits from Vaclav Kotesovec, Feb 19 2023

A245635 Decimal expansion of Integral_{x>=0} (1 / Gamma''(x)) dx.

Original entry on oeis.org

2, 2, 5, 2, 4, 4, 3, 9, 5, 2, 2, 2, 6, 5, 6, 1, 9, 6, 6, 6, 0, 8, 2, 2, 9, 9, 5, 8, 8, 7, 7, 9, 4, 2, 1, 3, 8, 4, 8, 6, 1, 6, 8, 5, 4, 6, 2, 0, 4, 2, 7, 4, 1, 9, 0, 1, 0, 8, 3, 0, 6, 1, 5, 2, 9, 6, 0, 2, 3, 1, 4, 3, 9, 2, 5, 0, 9, 9, 0, 9, 7, 3, 7, 4, 7, 1, 4, 1, 9, 5, 0, 9, 3, 6, 2, 2, 8, 3, 6, 9, 2, 7, 4, 2, 4

Offset: 1

Robert G. Wilson v, Jul 28 2014

			2.25244395222656196660822995887794213848616854620427419010830615296023143925099...

Cf. A058655.

RealDigits[ NIntegrate[ 1/Gamma''[x], {x, 0, Infinity}, AccuracyGoal -> 111, WorkingPrecision -> 111]][[1]]

Name corrected by Andrey Zabolotskiy, Dec 29 2023

A273051 Decimal expansion of the second moment of the reciprocal gamma distribution.

Original entry on oeis.org

4, 8, 3, 6, 4, 8, 5, 9, 7, 4, 6, 3, 3, 4, 2, 6, 8, 9, 4, 7, 3, 6, 3, 6, 0, 6, 9, 2, 3, 2, 1, 1, 3, 8, 9, 2, 4, 3, 6, 8, 5, 1, 6, 0, 8, 1, 0, 7, 3, 6, 0, 7, 2, 2, 9, 0, 3, 2, 9, 4, 2, 2, 4, 2, 1, 6, 0, 2, 7, 8, 6, 8, 4, 3, 7, 9, 7, 4, 5, 5, 2, 9, 5, 2, 3, 1, 3, 6, 1, 1, 0, 4, 0, 0, 3, 9, 3, 4, 4, 3, 7

Offset: 1

Jean-François Alcover, May 14 2016

			4.83648597463342689473636069232113892436851608107360722903294224216...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.6 Fransén-Robinson constant, p. 262.

Steven R. Finch Errata and Addenda to Mathematical Constants p. 35.
Eric Weisstein's MathWorld Fransén-Robinson Constant
Wikipedia, Reciprocal gamma function

Cf. A058655, A273017.

digits = 101;
I0 = NIntegrate[1/Gamma[x], {x, 0, Infinity}, WorkingPrecision -> digits + 5];
M2 = (1/I0) NIntegrate[x^2/Gamma[x], {x, 0, Infinity}, WorkingPrecision -> digits + 5];
RealDigits[M2, 10, digits][[1]]

default(realprecision, 120); intnum(x=0, [[1], 1], x^2/gamma(x))/intnum(x=0, [[1], 1], 1/gamma(x)) \\ Vaclav Kotesovec, May 14 2016

(1/I)*Integral_{x>=0} x^2/gamma(x) dx where I = Integral_{x>=0} 1/gamma(x) dx is the Fransén-Robinson constant.

A360809 Decimal expansion of the area under the curve of the reciprocal of the Luschny factorial function from zero to infinity.

Original entry on oeis.org

2, 5, 8, 6, 7, 0, 5, 0, 5, 9, 7, 8, 6, 8, 0, 8, 2, 2, 7, 7, 7, 8, 1, 0, 6, 8, 7, 2, 9, 4, 6, 9, 6, 0, 2, 1, 3, 5, 7, 3, 0, 9, 6, 2, 7, 4, 2, 4, 8, 9, 3, 6, 1, 2, 4, 4, 6, 7, 0, 8, 2, 4, 2, 2, 5, 8, 5, 9, 4, 0, 4, 5, 5, 6, 0, 6, 6, 4, 3, 4, 2, 6, 4, 2, 8, 8, 2, 7, 7, 7, 5, 6, 7, 5, 3, 9, 0, 8, 8, 7, 6, 4, 4, 6, 9, 9, 8, 1

Offset: 1

Hywel Normington, Feb 21 2023

			2.58670505978680822777810687294696021357309627424893612446708242258594...

Peter Luschny, Is the Gamma-function misdefined? Or: Hadamard vs Euler - Who found the better Gamma function?.
Hywel Normington, Python Code, 2023.
Hywel Normington, Reciprocal Gamma Integrals, 2023.

Cf. A360375, A058655.

L := proc(x) local G, S, y; if x = 0 then return 0.5 fi; y := x * 0.5; if is(x < 0) then y := -y fi; G := y * (Psi(y + 0.5) - Psi(y)) - 0.5; if is(x < 0) then return G/(-x)! fi; y := Pi * x; S := sin(y) / y; (1 - S * G) * x! end: RL := x -> 1 / L(x):
IntRL := n -> evalf[n](Int(RL, 0..n, method = Gquad)): IntRL(40); # _Peter Luschny, Feb 22 2023

RealDigits[NIntegrate[1 / (Gamma[x+1] * (1 - (x/2 * (PolyGamma[0, (x+1)/2] - PolyGamma[0, x/2]) - 1/2) * Sin[Pi*x]/(Pi*x))), {x, 0, Infinity}, WorkingPrecision -> 110, MaxRecursion -> Infinity]][[1]] (* Vaclav Kotesovec, Feb 22 2023 *)

default(realprecision, 500); intnum(x=0, [[1], 1], 1 / (gamma(x+1) * (1 - (x/2 * (psi((x+1)/2) - psi(x/2)) - 1/2) * sin(Pi*x)/(Pi*x)))) \\ (default(realprecision, 200) is enough for 59 valid digits, \p 500 for 102 valid digits, \p 1000 for 148 valid digits). - Vaclav Kotesovec, Feb 22 2023

L(x) = Gamma(x+1)P(x), where P(x) = 1 - g(x)*sin(Pi*x)/(Pi*x) and g(x) = (x/2)*(Psi((x+1)/2) - Psi(x/2)) - 1/2.

Equals Integral_{0..oo} 1/L(x) dx.

More digits from Vaclav Kotesovec, Feb 22 2023

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A229191 Decimal expansion of the integral_{x=0..Infinity} 1/x^x dx.

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A046943 Continued fraction for Fransen-Robinson constant Integral_{x>=0} 1/Gamma(x).

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A247377 Decimal expansion of Integral_{0..oo} 1/Gamma(1+x) dx, a variation of the Fransén-Robinson constant.

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A273017 Decimal expansion of the first moment of the reciprocal gamma distribution.

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A360375 Decimal expansion of the area under the curve of the reciprocal of the Hadamard gamma function from zero to infinity.

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A245635 Decimal expansion of Integral_{x>=0} (1 / Gamma''(x)) dx.

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A273051 Decimal expansion of the second moment of the reciprocal gamma distribution.

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