A094644 Continued fraction for e^gamma.

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

0,3

Comments

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).

References

G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 10.

J. Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 97.

Links

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.

Example

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

Mathematica

ContinuedFraction[ Exp[ EulerGamma], 100]

Prog

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

Crossrefs

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]

Sequence in context: A285175 A016599 A079650 * A113046 A384868 A245541

Adjacent sequences: A094641 A094642 A094643 * A094645 A094646 A094647

Keyword

nonn,cofr,easy

Author

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

Extensions

Offset changed by Andrew Howroyd, Aug 07 2024

Status

approved