A073233 -id:A073233 - cp's OEIS frontend

A178394 Decimal expansion of e!.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 27 2010

Gamma(e+1).

			4.2608204763570033817001212246457024649334243739593219749116...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
D. M. Bătinetu-Giurgiu, Problem 11808, The American Mathematical Monthly, Vol. 121, No. 10 (2014), p. 947; A Gamma Integral Limit, Solution to Problem 11808 by Roberto Tauraso, ibid., Vol. 123, No. 9 (2016), pp. 944-945.

Cf. A073233, A111293, A144749.

Mathematica
```
RealDigits[N[E!,200]]
```

Equals Integral_{x>=0} x^e/e^x dx. - Alois P. Heinz, Aug 27 2015

Equals lim_{n->oo} n^2 * Integral_{x=n!^(1/n)..(n+1)!^(1/(n+1))} Gamma(n*x) dx (Bătinetu-Giurgiu, 2014). - Amiram Eldar, Mar 26 2022

A236098 Decimal expansion of the real part of -Pi^(i*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jan 19 2014

The imaginary part is in A236099.

			0.898400579757743645668580370503151417548899949405011660808338878...

Stanislav Sykora, Table of n, a(n) for n = 0..10000

Cf. A000796, A053510, A073233, A231736, A236099, A236100, A236101.

RealDigits[Cos[Log[Pi]*Pi], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals -real(Pi^(i*Pi)) = -cos(Pi*log(Pi)) = -cos(A231736).

A236099 Decimal expansion of the imaginary part of -Pi^(I*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jan 19 2014

The real part is in A236098.

			0.4391769555554458943694547132383960901362100969781901495952341681...

Stanislav Sykora, Table of n, a(n) for n = 0..10000

Cf. A000796, A053510, A073233, A231736, A236098, A236100, A236101.

RealDigits[Im[-Pi^(I Pi)],10,120][[1]] (* Harvey P. Dale, Jan 02 2018 *)

-imag(Pi^(I*Pi)) = -sin(Pi*log(Pi)) = -sin(A231736).

A236100 Decimal expansion of the real part of Pi^(I/Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jan 19 2014

The imaginary part is in A236101.

			0.9343453036786376942622408604544211862401851213899337514...

Stanislav Sykora, Table of n, a(n) for n = 0..10000

Cf. A000796, A053510, A073233, A231736, A236101, A236098, A236099.

RealDigits[Re[Pi^(I/Pi)],10,120][[1]] (* Harvey P. Dale, Aug 02 2017 *)

real(Pi^(I/Pi)) = cos(log(Pi)/Pi) = cos(A231737).

A236101 Decimal expansion of the imaginary part of Pi^(i/Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Jan 19 2014

The real part is in A236100.

			0.35636898503331389990769183737865940578845872790065930829963...

Stanislav Sykora, Table of n, a(n) for n = 0..10000

Cf. A000796, A053510, A073233, A231736, A236100, A236098, A236099.

RealDigits[Sin[Log[Pi]/Pi], 10, 120][[1]] (* Amiram Eldar, Jun 06 2023 *)

Equals imag(Pi^(i/Pi)) = sin(log(Pi)/Pi) = sin(A231737).

A073239 Decimal expansion of (1/Pi)^Pi.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 25 2002

(1/Pi)^Pi = Pi^(-Pi) = 1/(Pi^Pi) (reciprocal of A073233).

			0.02742569312329810611955627085...

Cf. A000796 (Pi), A049541 (1/Pi), A073238 (Pi^(1/Pi)), A073240 ((1/Pi)^(1/Pi)), A073233 (Pi^Pi).

Join[{0},RealDigits[(1/Pi)^Pi,10,120][[1]]] (* Harvey P. Dale, Nov 30 2011 *)

PARI
```
(1/Pi)^Pi
```

A073237 a(n) = ceiling(Pi^Pi^...^Pi), where Pi appears n times.

Original entry on oeis.org

1, 4, 37, 1340164183006357436

Offset: 0

Rick L. Shepherd, Jul 25 2002

nonn

Decimal expansions (before taking ceiling) of Pi (A000796), Pi^Pi (A073233) and Pi^Pi^Pi (A073234) correspond to a(1), a(2) and a(3), respectively. See A073236 for same sequence rounded to nearest integer. This sequence is similar to A004002, which deals with e (but rounds).

a(4) has 666262452970848504 digits. - Martin Renner, Aug 19 2023

Cf. A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073236 (Pi^Pi^...^Pi, n times, rounded), A004002 (Benford numbers), A056072 (similar to A004002 but takes floor).

p:= n-> `if`(n=0, 1, Pi^p(n-1)):
a:= n-> ceil(p(n)):
seq(a(n), n=0..3);  # Alois P. Heinz, Jul 20 2024

p=0; for(n=0,3, p=Pi^p; print1(ceil(p),",")) \\ n=4 produces too large an exponent for PARI.

a(n) = ceiling(Pi^Pi^...^Pi), where Pi occurs n times, a(0) = 1 (=Pi^0).

A202953 Decimal expansion of x^x with x=Pi^Pi.

Original entry on oeis.org

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

Offset: 57

M. F. Hasler, Dec 26 2011

			887455172183124295874631455225434602688412866765466125005.158854842801290205563781894793427...

Cf. A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073235 ((Pi^Pi)^Pi), A202949 ((e^e)^(e^e)).

With[{x=Pi^Pi},RealDigits[x^x,10,120][[1]]] (* Harvey P. Dale, Dec 31 2011 *)

PARI
```
p(x)=x^x /* then type p(p(Pi)) */
```

A260634 Decimal expansion of 4^Pi.

Original entry on oeis.org

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

Offset: 2

Li GAN, Nov 11 2015

			77.880233648...

Cf. A217459 (2^Pi), A260629 (3^Pi), A073233, A260635.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^100*4^pi))); // Vincenzo Librandi, Nov 30 2015

First@ RealDigits@ N[4^Pi, 120] (* Michael De Vlieger, Nov 12 2015 *)

PARI
```
4^Pi
```

{ default(realprecision, 100); x=(4^Pi)/10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 29 2015

More digits from Altug Alkan, Nov 11 2015

A260635 Decimal expansion of 5^Pi.

Original entry on oeis.org

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

Offset: 3

Li GAN, Nov 11 2015

			156.99254530...

Cf. A217459 (2^Pi), A260629 (3^Pi), A073233, A260634.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^100*5^pi))); // Vincenzo Librandi, Nov 30 2015

First@ RealDigits@ N[5^Pi, 120] (* Michael De Vlieger, Nov 12 2015 *)

PARI
```
5^Pi
```

{ default(realprecision, 100); x=(5^Pi)/100; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 29 2015

More digits from Altug Alkan, Nov 11 2015

cp's OEIS Frontend

A178394 Decimal expansion of e!.

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A236098 Decimal expansion of the real part of -Pi^(i*Pi).

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A236099 Decimal expansion of the imaginary part of -Pi^(I*Pi).

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A236100 Decimal expansion of the real part of Pi^(I/Pi).

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A236101 Decimal expansion of the imaginary part of Pi^(i/Pi).

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A073239 Decimal expansion of (1/Pi)^Pi.

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PARI

A073237 a(n) = ceiling(Pi^Pi^...^Pi), where Pi appears n times.

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Maple

PARI

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A202953 Decimal expansion of x^x with x=Pi^Pi.

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