A058287 -id:A058287 - cp's OEIS frontend

A039661 Decimal expansion of exp(Pi).

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

Offset: 2

N. J. A. Sloane

e^Pi and Pi^e (A059850) differ by hardly 3% in magnitude. The determination of the inequality sign between them does not require their actual evaluation, the result being immediate from the basic facts Pi>e and log(x+1)0) yields log(Pi)

The formulas give e^Pi, not a(n). Note that e^Pi - Pi = 19.999099979...; that's why e^Pi and 20 + Pi have many common decimal digits. - M. F. Hasler, Oct 24 2009

e^Pi is transcendental, as proved by Gelfond. - Charles R Greathouse IV, May 07 2013

Nesterenko proves that this constant is algebraically independent of Pi and Gamma(1/4) over Q. - Charles R Greathouse IV, Nov 11 2013

Sum of the volumes of all even-dimensional unit spheres. - Paolo Xausa, Nov 14 2021

			23.1406926327792690...

L. Berggren, J. Borwein, and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 101.

Harry J. Smith, Table of n, a(n) for n = 2..20000
Francis Bischoff, The Volume of an Even Dimensional Ball (2024)
Bikash Chakraborty, A Visual Proof that Pi^e < e^Pi, arXiv:1806.03163 [math.HO], 2018.
D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37 (2000), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437-479. See Problem 7.
Fouad Nakhli, Proof without Words Pi^e < e^Pi, Mathematics Magazine, 60(3) (1987), pp. 165.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
Simon Plouffe, exp(Pi) to 5000 digits.
Arjun K. Rathie, Young Hee Geum, Eunyoung Lim, and Hwajoon Kim, A note on hypergeometric representation of a new mathematical constant, J. Appl. Math. & Informatics (2025) Vol. 43, No. 3, 795-804. See p. 797.
Arjun K. Rathie, Gradimir V. Milovanović, and Richard B. Paris, Hypergeometric representations of Gelfond's constant and its generalisations, Serbian Academy of Sciences and Arts (2021).
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Andrés Vallejo and Italo Bove, Which is greater: e^Pi or Pi^e? An unorthodox solution to a classic puzzle, arXiv:2309.10826 [physics.class-ph], 2023.
Eric Weisstein, Gelfond's Constant.
Wikipedia, Gelfond's constant.
OEIS Wiki, Gelfond's constant.
Index entries for transcendental numbers.

Cf. A059850 (Pi^e).

Cf. A058287 = contfrac(e^Pi), A058288 = contfrac(Pi^e).

RealDigits[N[E^Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

default(realprecision, 20080); x=exp(1)^Pi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b039661.txt", n, " ", d)); \\ Harry J. Smith, Apr 18 2009

A039661(n)=default(realprecision,n);exp(Pi)\10^(3-n)%10 \\ M. F. Hasler, Oct 24 2009

e^Pi = 32*Product_{j>=0} (u(j+1)/u(j))^(2^(-j+1)) where u(0)=1 and v(0)=1/sqrt(2); u(n+1) = u(n)/2 + v(n)/2, v(n+1) = sqrt(u(n)*v(n)) (deduced from Salamin algorithm for Pi). - Benoit Cloitre, Aug 14 2003
e^Pi = Sum_{k>=0} a(k)/k!/2^k where a(0)=1, a(1)=6 and a(n) = (40 - 4*n + n^2)*a(n-2) for n>=2 (from expansion of exp(6*asin(x)) at x=1/2). - Jaume Oliver Lafont, Oct 21 2009
exp(Pi) ~= log(Pi) + 7*Pi. - Alexander R. Povolotsky, Oct 24 2009
Equals Sum_{k>=0} Pi^k/k!. - Paolo Xausa, Nov 14 2021

A064107 Continued fraction quotients for e^e = 15.15426224... (A073226).

Original entry on oeis.org

15, 6, 2, 13, 1, 3, 6, 2, 1, 1, 5, 1, 1, 1, 9, 4, 1, 1, 1, 6, 7, 1, 2, 4, 1, 2, 2, 24, 1, 2, 4, 56, 1, 1, 2, 4, 1, 75, 1, 5, 1, 2, 2, 1, 137, 2, 2, 97, 3, 16, 1, 1, 1, 1, 3, 5, 12, 1, 1, 2, 1, 53, 1, 2, 5, 3, 2, 4, 1, 2, 1, 39, 1, 2, 1, 4, 1, 11, 1, 5, 5, 1, 4, 1, 17, 12, 4, 82, 1, 4, 6, 25, 3, 2, 3

Offset: 0

Labos Elemer, Sep 17 2001

It was conjectured (but remains unproved) that this sequence is infinite and aperiodic, but it is difficult to determine who first posed this problem. - Vladimir Reshetnikov, Apr 27 2013

			15.154262241479264189760430... = 15 + 1/(6 + 1/(2 + 1/(13 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 30 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Transcendental Number.
Wikipedia, List of unsolved problems in mathematics, Analysis.
Wikipedia, Irrational number, Open questions.

Cf. A058287, A058288, A073226 (decimal expansion), A159825.

with(numtheory): cfrac(evalf((exp(1))^(exp(1)),2560),256,'quotients');

ContinuedFraction[E^E,100] (* Harvey P. Dale, Sep 29 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(exp(1))); for (n=1, 20000, write("b064107.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 30 2009

Offset changed by Andrew Howroyd, Aug 05 2024

A320428 Continued fraction expansion of exp(Pi/4).

Original entry on oeis.org

2, 5, 5, 1, 3, 25, 1, 1, 17, 1, 3, 3, 1, 12, 1, 8, 5, 3, 1, 46, 3, 4, 12, 1, 5, 22, 3, 2, 1, 7, 4, 2, 1, 13, 13, 8, 1, 1, 3, 1, 1, 1, 2, 1, 11, 1, 5, 2, 1, 4, 7, 1, 71, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 4, 6, 1, 9, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 2, 1, 2, 10, 1, 19, 2, 2, 4, 1

Offset: 0

Grant T Sanderson, Aug 28 2019

This value arises naturally by taking the ratio of the volume of a unit 2n-dimensional ball to the volume of the 2n-dimensional cube containing it (with side length 2) and summing over all n.

Greg Egan, Puzzle in which this value arises naturally
Grant Sanderson and Brady Haran, Darts in Higher Dimensions, Numberphile video (2019)

Cf. A160510 (decimal expansion), A058287, A087299, A329912 (Engel expansion).

Mathematica
```
ContinuedFraction[Exp[Pi/4], 100]
```

contfrac(exp(Pi/4)) \\ Felix Fröhlich, Aug 28 2019

A116907 Continued fraction expansion for e^(-e) = 0.0659880358453125370767901875.

Original entry on oeis.org

0, 15, 6, 2, 13, 1, 3, 6, 2, 1, 1, 5, 1, 1, 1, 9, 4, 1, 1, 1, 6, 7, 1, 2, 4, 1, 2, 2, 24, 1, 2, 4, 56, 1, 1, 2, 4, 1, 75, 1, 5, 1, 2, 2, 1, 137, 2, 2, 97, 3, 16, 1, 1, 1, 1, 3, 5, 12, 1, 1, 2, 1, 53, 1, 2, 5, 3, 2, 4, 1, 2, 1, 39, 1, 2, 1, 4, 1, 11, 1, 5, 5, 1, 4, 1, 17, 12, 4, 82, 1, 4, 6, 25, 3, 2, 3, 39

Offset: 1

Jonathan Vos Post, Mar 16 2006

e^(-e) = (1/e)^e = 1/(e^e) = (reciprocal of A073226). e^(-e) = 0.0659880358453125370767901875... = 0 + 1/15+ 1/6+ 1/2+ 1/13+ 1/1+ 1/3+ 1/6+ 1/2+ ... See also: A073230 Decimal expansion of (1/e)^e. See also: A064107 Continued fraction quotients for e^e = 15.15426223. See also: A058287 Continued fraction for e^Pi. See also: A058288 Continued fraction expansion of Pi^e.

Cf. A064107, A058287, A058288, A073226.

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A039661 Decimal expansion of exp(Pi).

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A064107 Continued fraction quotients for e^e = 15.15426224... (A073226).

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A320428 Continued fraction expansion of exp(Pi/4).

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A116907 Continued fraction expansion for e^(-e) = 0.0659880358453125370767901875.

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