A094644 - OEIS

%I #63 Aug 07 2024 10:18:32

%S 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,

%T 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,

%U 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

%N Continued fraction for e^gamma.

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

%C 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).

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

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

%H T. D. Noe, <a href="/A094644/b094644.txt">Table of n, a(n) for n = 0..9999</a> (444 terms from Bo Gyu Jeong)

%H G. Caveney, J.-L. Nicolas, and J. Sondow, <a href="http://www.integers-ejcnt.org/l33/l33.pdf">Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis</a>, Integers 11 (2011), Article A33.

%H Jonathan Sondow, <a href="https://www.dropbox.com/s/2ewoxk79fz512py/AntiSymCE.pdf?dl=0">An antisymmetric formula for Euler's constant</a>, Math. Mag. 71 (1998), 219-220.

%H Jonathan Sondow, <a href="https://doi.org/10.1080/0025570X.1998.11996638">An antisymmetric formula for Euler's constant</a>, Math. Mag. 71 (1998), 219-220.

%H Jonathan Sondow, <a href="http://www.ams.org/journals/proc/2003-131-11/S0002-9939-03-07081-3/S0002-9939-03-07081-3.pdf">Criteria for irrationality of Euler's constant</a>, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.

%H Jonathan Sondow, <a href="http://arXiv.org/abs/math.CA/0211148">Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, arXiv:math/0211148 [math.CA], 2002-2004.

%H Jonathan Sondow, <a href="https://www.jstor.org/stable/30037385">Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, Amer. Math. Monthly 112 (2005), 61-65.

%H Jonathan Sondow, <a href="http://arXiv.org/abs/math.CA/0306008">An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma</a>, arXiv:math/0306008 [math.CA], 2003.

%H Jonathan Sondow, <a href="http://arXiv.org/abs/math.NT/0401406">A faster product for pi and a new integral for ln pi/2</a>, arXiv:math/0401406 [math.NT], 2004.

%H Jonathan Sondow, <a href="https://www.jstor.org/stable/30037575">A faster product for pi and a new integral for ln pi/2</a>, Amer. Math. Monthly 112 (2005), 729-734 and <a href="https://www.jstor.org/stable/27642026">113 (2006), 670</a>.

%H Jonathan Sondow and Sergey Zlobin, <a href="http://arXiv.org/abs/math.NT/0211075">A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant</a>, arXiv:math/0211075 [math.NT], 2002-2009.

%H Jonathan Sondow and Sergey Zlobin, <a href="https://doi.org/10.2478/s12175-009-0127-2">A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant</a>, Math. Slovaca 59 (2009), 1-8.

%H Jonathan Sondow and Wadim Zudilin, <a href="http://arXiv.org/abs/math.NT/0304021">Euler's constant, q-logarithms and formulas of Ramanujan and Gosper</a>,  arXiv:math/0304021 [math.NT], 2003.

%H Jonathan Sondow and Wadim Zudilin, <a href="https://doi.org/10.1007/s11139-006-0075-1">Euler's constant, q-logarithms and formulas of Ramanujan and Gosper</a>, Ramanujan J. 12 (2006), 225-244.

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

%t ContinuedFraction[ Exp[ EulerGamma], 100]

%o (PARI) contfrac(exp(Euler)) \\ _Amiram Eldar_, Jun 13 2021

%Y Cf. A073004 = decimal expansion of exp(gamma).

%Y Gamma is the Euler-Mascheroni constant A001620.

%Y Cf. A079650 = continued fraction for exp(-gamma). [From _R. J. Mathar_, Sep 05 2008]

%K nonn,cofr,easy

%O 0,3

%A _Jonathan Sondow_ and _Robert G. Wilson v_, May 18 2004

%E Offset changed by _Andrew Howroyd_, Aug 07 2024