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Exponential Factorial


The exponential factorial is defined by the recurrence relation

 a_n=n^(a_(n-1)),
(1)

where a_0=1. The first few terms are therefore

a_1=1
(2)
a_2=2^1=2
(3)
a_3=3^(2^1)=3^2=9
(4)
a_4=4^(3^(2^1))=4^9=262144
(5)

... (OEIS A049384). The term a_5=5^(262144) has 183231 digits.

The exponential factorial is therefore a kind of "factorial power tower."

The sum of the reciprocals of the exponential factorials is given by

S=sum_(k=1)^(infty)1/(a_k)
(6)
=1.61111492580837673611...11_()_(183213)272243682859...
(7)

(OEIS A080219). This sum is a Liouville number and is therefore transcendental.


See also

Liouville Number, Power Tower, Transcendental Number

This entry contributed by Jonathan Sondow (author's link)

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References

Sloane, N. J. A. Sequences A049384 and A080219 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Exponential Factorial

Cite this as:

Sondow, Jonathan. "Exponential Factorial." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExponentialFactorial.html

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