Hyperfactorial | |
---|---|

Notation | \(H(n)\) |

Type | Combinatorial |

Based on | Factorial |

Growth rate | \(f_{2}(n)\) |

Author | Sloane and Plouffe |

Year | 1995 |

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.- 2
^{108}is the largest known power of two not containing digit 9. - 666
^{108}is the first power of 666 larger than a centillion. - It is also considered sacred by the Dharmic religions.
- The FM broadcast band ends at 108 MHz 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 107.9 MHz. And the airband starts at the same frequency.
- It is the number of playing cards in an UNO deck, and the atomic number of the element hassium, which had the systematic symbol Uno.

- 2
**114**is the sum of the hyperfactorials of the first four nonnegative numbers.**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.

## Sources

## See also

**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