A073236 -id:A073236 - cp's OEIS frontend

A073233 Decimal expansion of Pi^Pi.

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

Offset: 2

Rick L. Shepherd, Jul 21 2002

A weak form of Schanuel's Conjecture implies that Pi^Pi is transcendental--see Marques and Sondow (2012).

			36.4621596072079117709908260226...

Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2012-2013.

Cf. A000796 (Pi), A073234 (Pi^Pi^Pi), A073237 (ceil(Pi^Pi^...^Pi), n Pi's), A073238 (Pi^(1/Pi)), A073239 ((1/Pi)^Pi), A073240 ((1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073236 (Pi analog of A004002).
Cf. A073226 (e^e).
Cf. A049006 (i^i), A116186 (real part of i^i^i).
Cf. A194555 (real part of i^e^Pi).

RealDigits[N[Pi^Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

PARI
```
Pi^Pi
```

{ default(realprecision, 20080); x=Pi^Pi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073233.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

A202955 Decimal expansion of Pi^Pi^Pi^Pi.

Original entry on oeis.org

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

Offset: 666262452970848504

M. F. Hasler, Dec 26 2011

The offset equals the floor(Pi^Pi^Pi*Log_10(Pi))+1. - Robert G. Wilson v, Mar 13 2014

Whether this number is an integer or not is an open question. It is also an open question whether Pi^Pi^Pi^...^Pi^Pi n times is an integer for any natural n > 4. - Eliora Ben-Gurion, Nov 17 2019

			9.080222455390617769723931713284287746516046358131897359946935926336845199... *10^666262452970848503

Cf. A073236, A085667, A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073235 ((Pi^Pi)^Pi), A202953 ((Pi^Pi)^(Pi^Pi)).

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[exp*Log10[base], nbrdgt + Floor[Log10[exp]] + 2]], 10, nbrdgt][[1]]; f[Pi, Pi^Pi^Pi] (* Robert G. Wilson v, Mar 13 2014 *)

LP(a,b)=[10^frac(a=log(a)/log(10)*b),a\1] /* returns [m,e] such that a^b = m*10^e */
LP(Pi,Pi^Pi^Pi)

A000796^A073234.

A073234 Decimal expansion of Pi^(Pi^Pi).

Original entry on oeis.org

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

Offset: 19

Rick L. Shepherd, Jul 25 2002

			1340164183006357435.29744912964...

Christopher Creutzig & Walter Oevel, MuPAD Tutorial. Berlin: Springer-Verlag (2004): 339.

Harry J. Smith, Table of n, a(n) for n = 19..20000

Cf. A000796 (Pi), A073233 (Pi^Pi), A202955 (Pi^Pi^Pi^Pi), A073236 (Pi^Pi^...^Pi, n times, rounded), A073235 ((Pi^Pi)^Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)).

RealDigits[Pi^Pi^Pi, 10, 100][[1]] (* Alonso del Arte, Jul 03 2012 *)

DIGITS := 100:
float(PI^(PI^PI)) // from Creutzig & Oevel (2004)

PARI
```
Pi^Pi^Pi
```

{ default(realprecision, 20100); x=Pi^Pi^Pi/10^18; for (n=19, 20000, d=floor(x); x=(x-d)*10; write("b073234.txt", n, " ", d)); } \\ Harry J. Smith, May 01 2009, corrected May 19 2009, had -n

A004002 Benford numbers: a(n) = e^e^...^e (n times) rounded to nearest integer.

Original entry on oeis.org

1, 3, 15, 3814279

Offset: 0

N. J. A. Sloane

nonn

The next term, a(4) ~ 2.3315*10^1656520, has 1656521 decimal digits and is therefore too large to be included. [Rephrased by M. F. Hasler, May 01 2013]

Named by Turner (1991) after the American electrical engineer and physicist Frank Albert Benford, Jr. (1883-1948). - Amiram Eldar, Jun 26 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. W. Clenshaw, F. W. J. Olver and P. R. Turner, Level-index arithmetic: An introductory survey in: P. R. Turner (ed.), Numerical Analysis and Parallel Processing, Lecture Notes in Mathematics, Vol. 1397, Springer, Berlin, Heidelberg, 1989, pp. 95-168.
Peter R. Turner, Will the "real" real arithmetic please stand up?, Notices Amer. Math. Soc., Vol. 38 (1991), pp. 298-304; entire issue.
Index entries for sequences related to Benford's law (The present sequence seems unrelated to Benford's law!)

Cf. A056072, A225053.

Cf. A073236. - Melissa O'Neill, Jul 04 2015

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

Round[NestList[Power[E, #] &, 1, 3]] (* Melissa O'Neill, Jul 04 2015 *)

a(n) = round(e^e^...^e), where e occurs n times, a(0) = 1 (= e^0). - Melissa O'Neill, Jul 04 2015

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).

A210508 a(n) = floor(((Pi^Pi)^Pi)^...^Pi).

Original entry on oeis.org

1, 3, 36, 80662, 2598761979625197, 2672496879634073073044276636697073994497970453122

Offset: 0

Jon Perry, Jan 25 2013

nonn

a(6) has more than 140 decimal digits and is too big to display. - L. Edson Jeffery, Jan 28 2013

			Pi^Pi = 36.4621596072079, so a(2) = 36.

Cf. A000796, A073236, A096232.

a:= n-> floor(Pi^(Pi^(n-1))):
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 20 2024

Floor[NestList[#^Pi &, Pi, 5]] (* Alonso del Arte, Jan 28 2013 *)

a(n)=Pi^Pi^n--\1 \\ Charles R Greathouse IV, Jan 28 2013

a(n) = floor(((Pi^Pi)^Pi)^...^Pi) (Pi appears n times).

a(n) = floor(Pi^(Pi^(n-1))). - Charles R Greathouse IV, Jan 28 2013

a(4) corrected by L. Edson Jeffery and Alonso del Arte, Jan 28 2013
a(5) added by L. Edson Jeffery, Jan 28 2013
a(0)=1 prepended by Alois P. Heinz, Jul 20 2024

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A073233 Decimal expansion of Pi^Pi.

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A202955 Decimal expansion of Pi^Pi^Pi^Pi.

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A073234 Decimal expansion of Pi^(Pi^Pi).

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A004002 Benford numbers: a(n) = e^e^...^e (n times) rounded to nearest integer.

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

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A210508 a(n) = floor(((Pi^Pi)^Pi)^...^Pi).

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