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

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

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

Examples

			23.1406926327792690...
		

References

• 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.

Links

• 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.

Crossrefs

Cf. A059850 (Pi^e).

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

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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