Hyperfactorial

The hyperfactorial is defined as \(H(n) = \prod^{n}_{i = 1} i^i = 1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdot \ldots \cdot n^n\).Hyperfactorial from Wolfram MathWorld
The first few values of \(H(n)\) for \(n = 1, 2, 3, 4, \ldots\) are 1, 4, 108, 27,648, 86,400,000, 4,031,078,400,000, 3,319,766,398,771,200,000, 55,696,437,941,726,556,979,200,000, 21,577,941,222,941,856,209,168,026,828,800,000, ... (). The sum of the reciprocals of these numbers is 2.2592954398150629..., which can be approximated as \(\sqrt[12]{17,688}\), or more precisely as \(\sqrt[7]{\sqrt[7]{3^{4}\cdot67\cdot3,929\cdot10,371,376,751}}\), a curious 18-decimal-place approximation where we have a double 7th root (7 is prime) of the product of seven prime factors.
\(H(n)\) has an FGH growth rate of very nearly \(f_2(f_2(n))\).
== Specific numbers ==
*108 is the third hyperfactorial number.
**2108 is the largest known power of two not containing digit 9.
**666108 is the first power of 666 larger than a centillion.
**It is also considered sacred by the .
**The ends at in most countries (except Japan, where the frequency range 99-108 MHz is reserved for digital broadcasting), but in most cases, the last usable carrier frequency is . And the starts at the same frequency.
**It is the number of s in an , and the of the element , which had the Uno.
*114 is the sum of the hyperfactorials of the first four nonnegative integers.
**It was also the PEGG value on May 20th, 2017.
**Its prime factorization is 2 × 3 × 19.
**The contains 114 s.
*27,648 is equal to four hyperfactorial \((1^1 \times 2^2 \times 3^3 \times 4^4)\). It is also equal to the number of square inches in a football goal.
*86,400,000 is equal to five hyperfactorial \((1^1 \times 2^2 \times 3^3 \times 4^4 \times 5^5)\). It is also equal to the number of milliseconds in a day.
== Approximations of these numbers ==
For 27,648:
Notation 	Lower bound 	Upper bound
Scientific notation 	\(2.7648\times10^4\)	\(2.7648\times10^4\)
Arrow notation 	\(30\uparrow3\) 	\(13\uparrow4\)
Steinhaus-Moser Notation 	5[3] 	6[3]
Copy notation 	2[5] 	3[5]
Chained arrow notation 	\(30\rightarrow3\) 	\(13\rightarrow4\)
Taro's multivariable Ackermann function	A(3,12) 	A(3,13)
Pound-Star Notation 	#*(20)*3 	#*(21)*3
PlantStar's Debut Notation 	[2] 	[3]
BEAF 	{30,3} 	{13,4}
Bashicu matrix system 	(0)[166] 	(0)[167]
Hyperfactorial array notation 	7! 	8!
Bird's array notation 	{30,3} 	{13,4}
Strong array notation 	s(30,3) 	s(13,4)
Fast-growing hierarchy 	\(f_{2}(11)\) 	\(f_{2}(12)\)
Hardy hierarchy 	\(H_{\omega^2}(11)\)	\(H_{\omega^2}(12)\)
Slow-growing hierarchy 	\(g_{\omega^3}(30)\)	\(g_{\omega^4}(13)\)
== Sources ==
== See also ==
ja:ハイパー階乗
Category:Factorials
Category:Functions