A079650 -id:A079650 - cp's OEIS frontend

A080130 Decimal expansion of exp(-gamma).

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

Offset: 0

Benoit Cloitre, Jan 26 2003

By Mertens's third theorem, lim_{k->oo} (H_{k-1}*Product_{prime p<=k} (1-1/p)) = exp(-gamma), where H_n is the n-th harmonic number. Let F(x) = lim_{n->oo} ((Sum_{k<=n} 1/k^x)*(Product_{prime p<=n} (1-1/p^x))) for real x in the interval 0 < x < 1. Consider the function F(s) of the complex variable s, but without the analytic continuation of the zeta function, in the critical strip 0 < Re(s) < 1. - Thomas Ordowski, Jan 26 2023

			0.56145948356688516982414321479088078676571...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Sections 1.5 p. 29, 2.7 p. 117 and 5.4 p. 285.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 202.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., 50 (2013), 527-628; arXiv:1303.1856 [math.NT], 2013.
Doron Zeilberger and Noam Zeilberger, Fractional Counting of Integer Partitions, 2018; Local copy [Pdf file only, no active links]

Cf. A000010, A000142, A001113, A001620, A002852, A007838, A073004, A079650, A322364, A322365, A322380, A322381.

R:= RealField(100); Exp(-EulerGamma(R)); // G. C. Greubel, Aug 28 2018

evalf(exp(-gamma), 120);  # Alois P. Heinz, Feb 24 2022

RealDigits[N[Exp[-EulerGamma], 200]][[1]] (* Arkadiusz Wesolowski, Aug 26 2012 *)

default(realprecision, 100); exp(-Euler) \\ G. C. Greubel, Aug 28 2018

Equals lim inf_{n->oo} phi(n)*log(log(n))/n. - Arkadiusz Wesolowski, Aug 26 2012
From Alois P. Heinz, Dec 05 2018: (Start)
Equals lim_{n->oo} A322364(n)/(n*A322365(n)).
Equals lim_{n->oo} A322380(n)/A322381(n). (End)
Equals lim_{k->oo} log(k)*Product_{prime p<=k} (1-1/p). - Amiram Eldar, Jul 09 2020
Equals lim_{n->oo} A007838(n)/A000142(n). - Alois P. Heinz, Feb 24 2022
Equals Product_{k>=1} (1+1/k)*exp(-1/k). - Amiram Eldar, Mar 20 2022
Equals A001113^(-A001620). - Omar E. Pol, Dec 14 2022
Equals lim_{n->oo} (A001008(p_n-1)/A002805(p_n-1))*(A038110(n+1)/A060753(n+1)), where p_n = A000040(n). - Thomas Ordowski, Jan 26 2023

A094644 Continued fraction for e^gamma.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 5, 4, 1, 1, 2, 2, 1, 7, 9, 1, 16, 1, 1, 1, 2, 6, 1, 2, 1, 6, 2, 59, 1, 1, 1, 3, 3, 3, 2, 1, 3, 5, 100, 1, 58, 1, 2, 1, 94, 1, 1, 2, 2, 10, 1, 2, 7, 1, 3, 4, 5, 3, 10, 1, 21, 1, 11, 1, 4, 1, 2, 2, 1, 2, 2, 1, 8, 3, 2, 1, 1, 6, 1, 2, 2, 1, 38, 2, 1, 4, 1, 3, 1, 1, 5, 3, 1, 52, 1, 2, 2, 1, 1

Offset: 0

Jonathan Sondow and Robert G. Wilson v, May 18 2004

Increasing partial quotients are: 1,3,5,7,9,16,59,100,129,314,2294,1568705

e^gamma appears in theorems of Mertens, Gronwall, Ramanujan, and Robin on primes, the sum-of-divisors function, and the Riemann Hypothesis (see Caveney-Nicolas-Sondow 2011, pp. 1-2).

			1 + 1/(1 + 1/(3 + 1/(1 + 1/(1 + 1/(3 + 1/(5 + 1/(4 + ...)))))))

J. Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 97.
G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 10.

T. D. Noe, Table of n, a(n) for n = 0..9999 (444 terms from Bo Gyu Jeong)
G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), Article A33.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
Jonathan 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.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma, arXiv:math/0306008 [math.CA], 2003.
Jonathan Sondow, A faster product for pi and a new integral for ln pi/2, arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Jonathan Sondow and Sergey Zlobin, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, arXiv:math/0211075 [math.NT], 2002-2009.
Jonathan Sondow and Sergey Zlobin, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant, Math. Slovaca 59 (2009), 1-8.
Jonathan Sondow and Wadim Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, arXiv:math/0304021 [math.NT], 2003.
Jonathan Sondow and Wadim Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, Ramanujan J. 12 (2006), 225-244.

Cf. A073004 = decimal expansion of exp(gamma).
Gamma is the Euler-Mascheroni constant A001620.
Cf. A079650 = continued fraction for exp(-gamma). [From R. J. Mathar, Sep 05 2008]

ContinuedFraction[ Exp[ EulerGamma], 100]

contfrac(exp(Euler)) \\ Amiram Eldar, Jun 13 2021

Offset changed by Andrew Howroyd, Aug 07 2024

A179950 Continued fraction for gamma^e, where gamma is the Euler-Mascheroni constant.

Original entry on oeis.org

0, 4, 2, 4, 1, 14, 2, 1, 4, 1, 19, 3, 1, 4, 10, 13, 1, 5, 67, 3, 1, 3, 1, 1, 1, 1, 3, 11, 1, 1, 5, 4, 3, 3, 1, 16, 1, 1, 1, 2, 3, 5, 1, 1, 41, 1, 1, 4, 17, 3, 24, 1, 7, 307, 8, 1, 1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 1, 1, 21, 3, 17, 2, 5, 5, 1, 1, 2, 1, 2, 915, 1, 1, 2, 4, 1, 1, 1, 3, 1, 19, 54, 3, 1, 9, 8

Offset: 0

Michel Lagneau, Aug 03 2010

			gamma^e =.224517251983232062665128293743... = 0 + 1/(4 + 1/(2 + 1/(4 + 1/(1 + ...))))

Cf. A001620 (gamma), A073018 (decimal expansion).

Cf. A094644, A079650, A073004.

with(numtheory): Digits := 300: convert(evalf(gamma^exp(1)), confrac);

A328339 Simple continued fraction expansion of Gamma(exp(gamma)+1).

Original entry on oeis.org

1, 1, 1, 1, 6, 15, 1, 4, 2, 1, 11, 2, 3, 1, 1, 13, 1, 1, 1, 1, 5, 1, 27, 1, 37, 1, 1, 2, 2, 2, 40, 1, 3, 1, 5, 1, 1, 3, 3942, 3, 2, 1, 13, 3, 11, 1, 2, 1, 15, 1, 1, 2, 1, 51, 37, 1, 1, 13, 4, 1, 1, 2, 5, 1, 1, 2, 1, 1, 5, 1, 75, 1, 16, 6, 2, 2, 1, 1, 7, 2, 4

Offset: 0

Daniel Hoyt, Oct 12 2019

'Gamma' is the gamma function, and 'gamma' is the Euler-Mascheroni constant.

Approximation of Gamma(exp(gamma)+1) in decimal: 1.6501566139104548059...

Cf. A094644, A073004, A079650.

ContinuedFraction[Gamma[1 + Exp[EulerGamma]], 81] (* Amiram Eldar, Oct 13 2019 *)

contfrac(gamma(exp(Euler)+1)) \\ Michel Marcus, Nov 04 2019

Gamma(exp(gamma)+1) = Gamma(exp(gamma)) * exp(gamma).

cp's OEIS Frontend

A080130 Decimal expansion of exp(-gamma).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A094644 Continued fraction for e^gamma.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A179950 Continued fraction for gamma^e, where gamma is the Euler-Mascheroni constant.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A328339 Simple continued fraction expansion of Gamma(exp(gamma)+1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula