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

1,3

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

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.

Links

Bo Gyu Jeong and T. D. Noe, Table of n, a(n) for n = 1..10000 (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 A245541 A209563

Adjacent sequences: A094641 A094642 A094643 * A094645 A094646 A094647

Keyword

nonn,cofr,easy

Author

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

Status

approved