Double exponential function

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This article is about double exponential functions. For the double exponential distribution, see Laplace distribution.

A double exponential function is a constant raised to the power of an exponential function. The general formula is $f(x) = a^{b^x}$ , which grows even faster than an exponential function. For example, if a = b = 10:

f(−1) ≈ 1.26
f(0) = 10
f(1) = 10¹⁰
f(2) = 10¹⁰⁰ = googol
f(3) = 10¹⁰⁰⁰
f(100) = 10^{10^100} = googolplex

Factorials grow faster than exponential functions, but much slower than double-exponential functions. Compare the hyper-exponential function, which grows even faster.

1 Doubly exponential sequences
2 Applications
3 See also
4 Further reading

[edit] Doubly exponential sequences

The nth value of each of the following integer sequences is proportional to a double exponential function of n. Although the sequences themselves are not double exponential functions, they can be defined using formulas in which such functions appear.

Fermat numbers

$F(m) = 2^{2^m}+1$

Double Mersenne numbers

$MM(p) = 2^{2^p-1}-1$

Elements of Sylvester's sequence (sequence A000058 in OEIS)

$s_n = \left\lfloor E^{2^{n+1}}+\frac12 \right\rfloor$

where E ≈ 1.26408 is Vardi's constant (sequence A076393 in OEIS).

Aho and Sloane (1973) investigate several other sequences that, like Sylvester's sequence, can be formed by rounding to the nearest integer the values of a doubly exponential function, and provide general conditions on the values of a sequence that cause it to be of this type.