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Hyperfactorial
Notation\(H(n)\)
TypeCombinatorial
Based onFactorial
Growth rate\(f_{2}(n)\)
AuthorSloane and Plouffe
Year1995
The hyperfactorial is defined as \(H(n) = \prod^{n}_{i = 1} i^i = 1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdot \ldots \cdot n^n\).[1]

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, ... (OEIS A002109). 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.
  • 114 is the sum of the hyperfactorials of the first four nonnegative numbers.
    • It was also the PEGG value on May 20th, 2017.
    • Its prime factorization is 2 × 3 × 19.
    • The Quran contains 114 surahs.
  • 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 \(2.7648\times10^4\)
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

  1. Hyperfactorial from Wolfram MathWorld

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
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial
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