| 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.
- 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 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.
- 114 is the sum of the hyperfactorials of the first four nonnegative integers.
- 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[]
See also[]
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: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial