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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 integers.
• 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 $$g_{\omega^3}(30)$$ $$g_{\omega^4}(13)$$