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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, 262,144, ... (OEIS A049384). The next number, 5262,144, has 183,231 digits and starts with 6,206,069,878,660,874,470,748,320,557,284,67... Exponential factorial of 6 is approximately \(10^{4.829261036 \cdot 10^{183,230}}\) and starts with 1,103,560,225,917,696,632,179,145,334,475,34....

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 1/262,144 have finite decimal expansions, and the reciprocal of 5262,144 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, the expofactorial of n can be written as n!1.

Sources

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

See also

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 · Primorial · Compositorial · Semiprimorial
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 · Bouncing Factorial