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A094644 Continued fraction for e^gamma. 6
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 (list; graph; refs; listen; history; text; internal format)
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.

J. Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.

J. Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.

J. 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.

J. Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma.

J. 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.

J. Sondow, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin, Math. Slovaca 59 (2009), 1-8.

J. Sondow and W. 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]

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

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Last modified October 14 14:29 EDT 2019. Contains 328018 sequences. (Running on oeis4.)