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A039661 Decimal expansion of exp(Pi). 29
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

e^pi and pi^e (A059850) differ hardly by 3% in magnitude. The determination of the inequality sign between them dispenses with their actual evaluation, the result being immediate from the basic facts pi>e and ln(x+1)<x for positive x, whence setting x=(pi/e)-1 (>0) yields ln(pi)<pi/e, or pi^e < e^pi.

The formulae 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. [From 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

REFERENCES

L. Berggren, J. Borwein and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422

LINKS

Harry J. Smith, Table of n, a(n) for n = 2..20000

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.

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

Eric Weisstein, Gelfond's Constant

Wikipedia, Gelfond's constant

OEIS Wiki, Gelfond's constant

FORMULA

32*prod(j>=0, (u(j+1)/u(j))^2^(-j+1)) where u(0)=1 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

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) [From Jaume Oliver Lafont, Oct 21 2009]

exp(Pi) ~= ln(Pi) + 7*Pi [From Alexander R. Povolotsky, Oct 24 2009]

EXAMPLE

23.1406926327792690...

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A127412 A152832 A211343 * A214684 A081877 A049076

Adjacent sequences:  A039658 A039659 A039660 * A039662 A039663 A039664

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Fixed my PARI program, had -n Harry J. Smith, May 19 2009

OEIS Wiki link by Daniel Forgues, Oct 02 2011

STATUS

approved

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Last modified November 1 00:50 EDT 2014. Contains 248878 sequences.