About: Exponential factorial   Sponge Permalink

An Entity of Type : owl:Thing, within Data Space : 134.155.108.49:8890 associated with source dataset(s)

The exponential factorial or expofactorial 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.... In Hyperfactorial array notation, expofactorial is equal to n!1.

AttributesValues
rdfs:label
  • Exponential factorial
rdfs:comment
  • The exponential factorial or expofactorial 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.... In Hyperfactorial array notation, expofactorial is equal to n!1.
sameAs
dcterms:subject
dbkwik:googology/p...iPageUsesTemplate
abstract
  • The exponential factorial or expofactorial 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.
Alternative Linked Data Views: ODE     Raw Data in: CXML | CSV | RDF ( N-Triples N3/Turtle JSON XML ) | OData ( Atom JSON ) | Microdata ( JSON HTML) | JSON-LD    About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 07.20.3217, on Linux (x86_64-pc-linux-gnu), Standard Edition
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2012 OpenLink Software