A085667 -id:A085667 - cp's OEIS frontend

A056072 a(n) = floor(e^e^ ... ^e), with n e's.

1, 2, 15, 3814279

Offset: 0

Robert G. Wilson v, Jul 26 2000

nonn

The next term is too large to include.
From Vladimir Reshetnikov, Apr 27 2013: (Start)
a(4) = 2331504399007195462289689911...2579139884667434294745087021 (1656521 decimal digits in total), given by initial segment of A085667.
a(5) has more than 10^10^6 decimal digits.
a(6) has more than 10^10^10^6 decimal digits. (End)

Cf. A001113, A004002, A003417, A064107, A159825, A073226, A073227, A085667, A096232, A225064, A225053.

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

Floor[NestList[Exp, 1, 3]] (* Vladimir Reshetnikov, Apr 29 2013 *)

A056072(n,f=floor)=f(exp(if(n>0,A056072(n-1,x->x))))  \\ M. F. Hasler, May 01 2013

A225064 Decimal expansion of the fractional part of e^e^e^e.

Original entry on oeis.org

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

Offset: 0

Vladimir Reshetnikov, Apr 26 2013

It was conjectured (but remains unproved) that this sequence is infinite and aperiodic.

			frac(e^e^e^e) = 0.2212029997921176538924919342....

Eric Weisstein's World of Mathematics, e

Cf. A085667 (includes integer part).

base = 10; terms = 120; First[RealDigits[FractionalPart[E^E^E^E], base, terms]]

a(n) = A085667(n+1656521), where 1656521 is the length of the integer part of e^e^e^e.

Offset corrected by Rick L. Shepherd, Jan 01 2014

A202949 Decimal expansion of (e^e)^(e^e), where e=exp(1).

Original entry on oeis.org

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

Offset: 18

M. F. Hasler, Dec 26 2011

			776486517915808457.38262707214480111269813738740893733361098023776562998338874696481792585472289...

Cf. A073226, A073227, A073228, A085667, A181180, A073232. - M. F. Hasler, Dec 26 2011

RealDigits[#^#&/@(E^E),10,120][[1]] (* Harvey P. Dale, Aug 31 2023 *)

default(realprecision,99); t=exp(1); t=t^t; t=t^t

A073226^A073226.

A361100 Decimal expansion of 2^(2^(2^(2^2))) = 2^^5.

Original entry on oeis.org

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

Offset: 19730

Marco Ripà, Mar 03 2023

2^0 = 1, 2^1 = 2, 2^2 = 4, 2^2^2 = 2^^3 = (2^2)^2 = 16,
2^2^2^2 = 2^^4 = (((2^2)^2)^2)^2 = 65536
so that 2^2^2^2^2 = 2^^5 = 2^(2^(2^(2^2))) = 2^65536 = 20035299304068464649790723515602557504478254755697514192650169737108940595563...

			2003529930406846464979072351560255750447825475569751419265016973710894059556311453089506130880933348
(...19529 digits omitted...)
5775699146577530041384717124577965048175856395072895337539755822087777506072339445587895905719156736.
The above example line shows the first one hundred decimal digits and the last one hundred digits with the number of unrepresented digits in parentheses.

Chai Wah Wu, Table of n, a(n) for n = 19730..39458
Googology Wiki, Tetration.
Pointless Large numbers stuff by Cookiefonster, 2.02 Knuth's Up-Arrows and the Hyper-Operators.
Robert P. Munafo, Sequence A094358, 2^^N = 1 mod N.

Cf. A085667, A014221, A241291, A241292, A241293, A241294, A241295, A241296, A241297, A241298, A241299, A243913.

nbrdgt = 100; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[ exp*Log10[ base], nbrdgt + Floor[ Log10[ exp]] + 2]], 10, nbrdgt][[1]]; f[2, 2^2^2^2]
IntegerDigits[2^65536][[;;100]] (* Paolo Xausa, Jan 31 2024 *)

def A361100(n): return (1<<(1<<(1<<(1<<(1<<1)))))//10**(39458-n)%10 # Chai Wah Wu, Apr 03 2023

Equals 2^2^2^2^2 = 2^^5 = (((((((((((((((2^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2)^2.

A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

Original entry on oeis.org

1, 2, 3, 16, 3814280

Offset: 1

Alexis Monnerot-Dumaine, Jan 21 2010

nonn

Log(a(1)) is defined if a(1) > 0, so a(1) = 1.
Log(log(a(2))) is defined if log(a(2)) > 0 => a(2) > 1 => a(2) = 2.
The sequence grows rapidly: a(6) = 2.33150...10^1656520, and is too large to include here.

			a(2) = 2 because log(log(2)) is defined and log(log(1)) is not;
a(3) = 3 because log(log(log(3))) is defined;
a(4) = 16 because log(log(log(log(16)))) is defined.
From _Robert G. Wilson v_, Jul 05 2022: (Start)
a(3) = ceiling(A001113).
a(4) = ceiling(A073226).
a(5) = ceiling(A073227).
a(6) = ceiling(A085667). (End)

Cf. A074785 (log(log(2))), A085667, A004002, A001113, A073226, A073227, A085667.

a(n) = my(k=1); while(1, my(s=k, i=0); while(s > 0, s=log(s); if(s > 0, i++)); if(i==n-1, return(k)); k++) \\ Felix Fröhlich, Nov 22 2015

For n > 2, a(n) = ceiling(e^(e^(...))) where e appears n-2 times.

cp's OEIS Frontend

A056072 a(n) = floor(e^e^ ... ^e), with n e's.

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A225064 Decimal expansion of the fractional part of e^e^e^e.

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A202949 Decimal expansion of (e^e)^(e^e), where e=exp(1).

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A361100 Decimal expansion of 2^(2^(2^(2^2))) = 2^^5.

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A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

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