A059850 -id:A059850 - 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

A073244 Decimal expansion of Pi - e.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 21 2002

			0.42331082513074800310235591192...

Cf. A059742 (Pi+e), A000796 (Pi), A001113 (e), A019609 (Pi*e), A061382 (Pi/e), A061360 (e/Pi), A039661 (e^Pi), A059850 (Pi^e), A073233 (Pi^Pi), A073226 (e^e), A049006 (i^i = e^(-Pi/2)).

Cf. A110564 for continued fraction for Pi - e.

RealDigits[N[Pi-E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI
```
Pi-exp(1)
```

A063504 Decimal expansion of e^Pi - Pi^e.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jul 30 2001

A classic calculus analysis problem is to discover whether e^Pi or Pi^e is the greater without the use of a calculator.

			0.681534914418223532301934163404812352676791108603519744242043855457416... - _Harry J. Smith_, Aug 24 2009

Paul J. Nahin, When Least Is Best, How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible, Princeton University Press, Princeton NJ, 2004, Page 144.
Alfred S. Posamentier & Ingmar Hehmann, Pi: A Biography of the World's Most Mysterious Number, Prometheus Books, NY 2002, pages 146, 301-304.

Harry J. Smith, Table of n, a(n) for n = 0..20000

Equals A039661 - A059850.

Cf. A063503.

Mathematica
```
RealDigits[N[E^Pi - Pi^E, 100]][[1]]
```

{ default(realprecision, 20080); e=exp(1); x=10*(e^Pi - Pi^e); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b063504.txt", n, " ", d)) } \\ Harry J. Smith, Aug 24 2009

Offset corrected by R. J. Mathar, Feb 05 2009

A058288 Continued fraction expansion of Pi^e.

Original entry on oeis.org

22, 2, 5, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 9, 15, 25, 1, 1, 5, 4, 1, 2, 1, 1, 50, 1, 1, 1, 1, 7, 1, 1, 1, 3, 6, 1, 20, 10, 1, 2, 10, 1, 8, 2, 2, 1, 1, 1, 4, 1, 43, 2, 2, 3, 1, 2, 8, 1, 1, 16, 1, 4, 1, 3, 1, 1, 1, 2, 1, 1, 6, 1, 2, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 9, 1, 1, 105, 1, 3, 6, 2, 1, 1, 3, 1, 3, 2, 1, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

			Pi^e = 22.459157718361045473... = 22 + 1/(2 + 1/(5 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 19 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A059850.

cfrac(evalf((evalf(Pi))^(exp(1)),2560),256,'quotients');

Mathematica
```
ContinuedFraction[ Pi^E, 100]
```
PARI
```
contfrac(Pi^exp(1))
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^exp(1)); for (n=0, 20000, write("b058288.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Apr 19 2009

More terms from Jason Earls, Jul 12 2001

A104691 Decimal expansions of e and Pi interlaced.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Apr 23 2005

Within 2.5 per cent of e^Pi (A039661: 23.14069...) or 5.5 per cent of Pi^e (A059850: 22.45915...). - Robert G. Wilson v, Jan 04 2013

Cf. A001355.

Riffle[RealDigits[E, 10, 53][[1]], RealDigits[Pi, 10, 53][[1]]] (* Robert G. Wilson v, Jan 04 2013 *)

A038153 Beatty sequence for Pi^e.

Original entry on oeis.org

22, 44, 67, 89, 112, 134, 157, 179, 202, 224, 247, 269, 291, 314, 336, 359, 381, 404, 426, 449, 471, 494, 516, 539, 561, 583, 606, 628, 651, 673, 696, 718, 741, 763, 786, 808, 830, 853, 875, 898, 920, 943, 965, 988, 1010, 1033, 1055, 1078, 1100, 1122, 1145

Offset: 1

Felice Russo

nonn

Karl V. Keller, Jr., Table of n, a(n) for n = 1..1000
Index entries for sequences related to Beatty sequences

Cf. A059850. - R. J. Mathar, Oct 10 2010

A038153 := proc(n) floor(n*Pi^exp(1)) ; end proc: seq(A038153(n),n=1..80) ; # R. J. Mathar, Oct 10 2010

Floor[Range[100]*Pi^E] (* Paolo Xausa, Jul 05 2024 *)

a(n) = floor(n * 22.4591577...).

Extended by R. J. Mathar, Oct 10 2010

A059197 Engel expansion of Pi^e = 22.4592.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 8, 17, 111, 236, 419, 2475, 3741, 4123, 5563, 5622, 18000, 33641, 42744, 130605, 696987, 975174, 1034590, 2806140, 14026897, 14137435, 65788323, 73121589, 229261119

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1022
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A059850.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Pi^E, 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A092171 Decimal expansion of Pi^(-e).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 01 2004

			0.0445252672669229...

Equals 1/A059850.

A092173 Decimal expansion of Pi^(2*e).

Original entry on oeis.org

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

Offset: 3

Mohammad K. Azarian, Apr 01 2004

			504.41376541821652...

Cf. A059850.

RealDigits[Pi^(2E),10,120][[1]] (* Harvey P. Dale, May 12 2025 *)

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cp's OEIS Frontend

A063503 Continued fraction for e^Pi - Pi^e (A063504 = A039661 - A059850).

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A039661 Decimal expansion of exp(Pi).

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A073244 Decimal expansion of Pi - e.

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A063504 Decimal expansion of e^Pi - Pi^e.

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A058288 Continued fraction expansion of Pi^e.

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A104691 Decimal expansions of e and Pi interlaced.

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A038153 Beatty sequence for Pi^e.

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