

A141606


Decimal expansion of (W(e1)/(e1))^(1/(1e)), where W(z) denotes the Lambert W function and e = 2.718281828...


4



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

1,2


COMMENTS

Solution for x in x^(x^(e1)) = e.
(W((y1)log(z))/((y1)log(z)))^(1/(1y)) = e^(W((y1)log(z))/(y1)) so that (W(e1)/(e1))^(1/(1e)) = e^(W(e1)/(e1)).  Ross La Haye, Aug 27 2008
Consider the expression x^x^x^x... where x appears y times. For, say, y = 4 this type of expression is conventionally evaluated as if bracketed x^(x^(x^x)) and is referred to as a "power tower". However, we can also bracket x^x^x^x from the bottom up, e.g., (x^x)^x)^x = x^(x^3). In general, this bracketing will simplify x^x^x^x... to x^(x^(y1)) when x appears y times in the expression. Solving the equation x^(x^(y1)) = z for x gives x = (W((y1)log(z))/((y1)log(z)))^(1/(1y)). And setting y = z = e gives the result indicated by this sequence. Special thanks are due to Mike Wentz for introducing me to the "bottom up" bracketing of x^x^x^x... and the motivation for its investigation.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Lambert WFunction
Eric Weisstein, Power Tower


EXAMPLE

1.57844691419127618691147145725058871862508588172697263709178296257...


MATHEMATICA

RealDigits[(ProductLog[E1]/(E1))^(1/(1E)), 10, 111][[1]]


PROG

(PARI) (lambertw(exp(1)1)/(exp(1)1))^(1/(1exp(1))) \\ G. C. Greubel, Mar 02 2018


CROSSREFS

Cf. A001113.
Cf. A143913, A143914, A143915.  Ross La Haye, Sep 05 2008
Sequence in context: A245278 A155855 A070366 * A197491 A254274 A068001
Adjacent sequences: A141603 A141604 A141605 * A141607 A141608 A141609


KEYWORD

cons,nonn


AUTHOR

Ross La Haye, Aug 21 2008, Aug 26 2008


EXTENSIONS

More terms from Robert G. Wilson v, Aug 25 2008


STATUS

approved



