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A039661
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Decimal expansion of exp(Pi).
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28
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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
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OFFSET
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2,1
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COMMENTS
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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 proven by Gelfond. - Charles R Greathouse IV, May 07 2013
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REFERENCES
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L. Berggren, J. Borwein and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422
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LINKS
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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.
_Simon Plouffe_, exp(pi) to 5000 digits
Eric Weisstein. Gelfond's Constant
Wikipedia, Gelfond's Constant.
OEIS Wiki, Gelfond's constant
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FORMULA
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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]
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EXAMPLE
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23.1406926327792690...
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MATHEMATICA
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RealDigits[N[E^Pi, 200]] [From Vladimir Joseph Stephan Orlovsky, May 27 2010]
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PROG
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(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)); } [From Harry J. Smith, Apr 18 2009]
(PARI) A039661(n)=default(realprecision, n); exp(Pi)\10^(3-n)%10 [From M. F. Hasler, Oct 24 2009]
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CROSSREFS
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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
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KEYWORD
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nonn,cons
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Fixed my PARI program, had -n Harry J. Smith, May 19 2009
OEIS Wiki link by Daniel Forgues, Oct 02 2011
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STATUS
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approved
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