

A039661


Decimal expansion of exp(Pi).


34



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

2,1


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)<x for positive x, whence setting x=(Pi/e)1 (>0) yields log(Pi)<Pi/e, or Pi^e < e^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


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
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), 407436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437479. 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. 13191348. (English translation)
Simon Plouffe, exp(pi) to 5000 digits
Arjun K. Rathie, Gradimir V. MilovanoviÄ‡, 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), 114116.
Eric Weisstein, Gelfond's Constant
Wikipedia, Gelfond's constant
OEIS Wiki, Gelfond's constant
Index entries for transcendental numbers


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(n2) 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


EXAMPLE

23.1406926327792690...


MATHEMATICA

RealDigits[N[E^Pi, 200]] (* 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=(xd)*10; write("b039661.txt", n, " ", d)); \\ Harry J. Smith, Apr 18 2009
(PARI) A039661(n)=default(realprecision, n); exp(Pi)\10^(3n)%10 \\ M. F. Hasler, Oct 24 2009


CROSSREFS

Cf. A059850 (Pi^e).
Cf. A058287 = contfrac(e^Pi), A058288 = contfrac(Pi^e).
Sequence in context: A306646 A152832 A211343 * A293668 A214684 A268727
Adjacent sequences: A039658 A039659 A039660 * A039662 A039663 A039664


KEYWORD

nonn,cons


AUTHOR

N. J. A. Sloane


STATUS

approved



