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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.

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.

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