A064107 -id:A064107 - 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

A159825 Continued fraction for e^e^e A073227.

Original entry on oeis.org

3814279, 9, 1, 1, 4, 1, 53, 26, 1, 13, 3, 1, 1, 22, 1, 226, 1, 5, 2, 1, 6, 2, 3, 1, 4, 1, 6, 39, 2, 1, 3, 1, 5, 1, 4, 1, 3, 1, 4, 1, 1, 19, 1, 2, 8899, 5, 2, 2, 1, 3, 3, 2, 2, 2, 1, 1, 3, 5, 1, 6, 10, 2, 1, 2, 1, 1, 1, 2, 2, 4, 1, 10, 2, 6, 1, 5, 6, 2, 4, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 11, 7, 3, 1, 4, 4

Offset: 0

Harry J. Smith, Apr 30 2009

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

			3814279.104760220592209... = 3814279 + 1/(9 + 1/(1 + 1/(1 + 1/(4 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Transcendental Number.
Wikipedia, Irrational number, Open questions.

Cf. A003417, A064107.

ContinuedFraction[E^E^E, 96] (* Vladimir Reshetnikov, Apr 27 2013 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(exp(exp(1)))); for (n=1, 20001, write("b159825.txt", n-1, " ", x[n])); }

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]]

A116907 Continued fraction expansion for e^(-e) = 0.0659880358453125370767901875.

Original entry on oeis.org

0, 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, 39

Offset: 1

Jonathan Vos Post, Mar 16 2006

e^(-e) = (1/e)^e = 1/(e^e) = (reciprocal of A073226). e^(-e) = 0.0659880358453125370767901875... = 0 + 1/15+ 1/6+ 1/2+ 1/13+ 1/1+ 1/3+ 1/6+ 1/2+ ... See also: A073230 Decimal expansion of (1/e)^e. See also: A064107 Continued fraction quotients for e^e = 15.15426223. See also: A058287 Continued fraction for e^Pi. See also: A058288 Continued fraction expansion of Pi^e.

Cf. A064107, A058287, A058288, A073226.

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]

cp's OEIS Frontend

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

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A159825 Continued fraction for e^e^e A073227.

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

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A116907 Continued fraction expansion for e^(-e) = 0.0659880358453125370767901875.

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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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