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Exponential factorial

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The exponential factorial[1] or expofactorial[2] is an exponential version of the factorial, recursively defined as \(a_0 = 1\) and \(a_n = n^{a_{n - 1}}\). For example, \(a_6 = 6^{5^{4^{3^{2^1}}}}\).

The first few \(a_n\) for \(n = 0, 1, 2, 3, \ldots\) are 1, 1, 2, 9, 262144, ... (OEIS A049384). The next number, 5262144, has 183231 digits and starts with 620606987866087447074832055728467... Exponential factorial of 6 is approximately \(10^{4.829261036 \cdot 10^{183230}}\) and starts with 110356022591769663217914533447534....

The sum of the reciprocals of these numbers is 2.6111149258083767361111...(183,213 1's)...1111272243... The long string of 1's appear because 1/9 = 0.111111111... and 1/2 and 1/262144 have finite decimal expansions, and the reciprocal of 5262144 is so small that more than 100,000 of the first decimal digits are zeroes.

The exponential factorial satisfies the bound \(a_n \leq {^{n-1}n}\) (tetration), and satisfies \(a_n < {^{n-k}n}\), for sufficiently large n (and any k).

In Hyperfactorial array notation, expofactorial is equal to n!1.

Sources Edit

  1. Exponential Factorial from Wolfram MathWorld
  2. A049384 from OEIS

See also Edit

Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial

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