A159825 -id:A159825 - 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

A064107 Continued fraction quotients for e^e = 15.15426224... (A073226).

Original entry on oeis.org

15, 6, 2, 13, 1, 3, 6, 2, 1, 1, 5, 1, 1, 1, 9, 4, 1, 1, 1, 6, 7, 1, 2, 4, 1, 2, 2, 24, 1, 2, 4, 56, 1, 1, 2, 4, 1, 75, 1, 5, 1, 2, 2, 1, 137, 2, 2, 97, 3, 16, 1, 1, 1, 1, 3, 5, 12, 1, 1, 2, 1, 53, 1, 2, 5, 3, 2, 4, 1, 2, 1, 39, 1, 2, 1, 4, 1, 11, 1, 5, 5, 1, 4, 1, 17, 12, 4, 82, 1, 4, 6, 25, 3, 2, 3

Offset: 0

Labos Elemer, Sep 17 2001

It was conjectured (but remains unproved) that this sequence is infinite and aperiodic, but it is difficult to determine who first posed this problem. - Vladimir Reshetnikov, Apr 27 2013

			15.154262241479264189760430... = 15 + 1/(6 + 1/(2 + 1/(13 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 30 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Transcendental Number.
Wikipedia, List of unsolved problems in mathematics, Analysis.
Wikipedia, Irrational number, Open questions.

Cf. A058287, A058288, A073226 (decimal expansion), A159825.

with(numtheory): cfrac(evalf((exp(1))^(exp(1)),2560),256,'quotients');

ContinuedFraction[E^E,100] (* Harvey P. Dale, Sep 29 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(exp(1))); for (n=1, 20000, write("b064107.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 30 2009

Offset changed by Andrew Howroyd, Aug 05 2024

A225053 Second terms of continued fractions for power towers e, e^e, e^e^e, ...

Original entry on oeis.org

1, 6, 9, 4

Offset: 1

Vladimir Reshetnikov, Apr 25 2013

It was conjectured (but remains unproved) that none of the power towers e, e^e, e^e^e, ... are integers. If so, the corresponding continued fractions contain at least 2 terms. If the conjecture fails, let the corresponding a(n) = 0.

			a(3) = 9 because floor(1/frac(e^e^e)) = 9, since e^e^e ~ 3814279.10476.

Eric Weisstein's World of Mathematics, e.
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration, Open questions

Cf. A003417, A064107, A159825, A225064, A004002.

A056072 yields the first term of the continued fraction.

$MaxExtraPrecision = Infinity; terms = 4; Map[Function[x, ContinuedFraction[x, 2][[2]]], NestList[Exp, E, terms - 1]]

A225062 Continued fraction for 1/frac(e^e^e^e). Also, continued fraction for e^e^e^e starting from the 2nd term.

Original entry on oeis.org

4, 1, 1, 11, 1, 1, 3, 1, 6, 2, 1, 3, 1, 1, 8, 1, 8, 2, 3, 1, 3, 3, 1, 1, 4, 22, 4, 2, 2, 4, 6, 1, 98, 1, 3, 1, 3, 1, 1, 3, 3, 1, 1, 1, 9, 2, 16, 1, 1, 1, 3, 3, 1, 11, 2, 1, 2, 1, 2, 5, 1, 11, 1, 7, 4, 1, 4, 12, 8, 1, 6, 1, 1, 1, 1, 4, 2, 2, 3, 2, 1, 1, 7, 1, 8, 8, 1, 117, 4, 6, 3, 1, 3, 1, 1, 4, 2, 2, 7, 1, 2, 1, 1, 3, 21, 1, 9, 6, 1, 1, 4, 2, 2, 1, 5

Offset: 1

Vladimir Reshetnikov, Apr 26 2013

The 1st term of continued fraction for e^e^e^e has 1656521 decimal digits, so it is not included in the sequence.

Eric Weisstein's World of Mathematics, e
Eric Weisstein's World of Mathematics, Power Tower

Cf. A003417, A064107, A159825.

$MaxExtraPrecision = Infinity; terms = 115; ContinuedFraction[1/FractionalPart[E^E^E^E], terms]

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

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A064107 Continued fraction quotients for e^e = 15.15426224... (A073226).

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A225053 Second terms of continued fractions for power towers e, e^e, e^e^e, ...

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A225062 Continued fraction for 1/frac(e^e^e^e). Also, continued fraction for e^e^e^e starting from the 2nd term.

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