The first few values of (known as a superfactorial) for , 2, ... are given by 1, 2, 12, 288, 34560, 24883200, ... (OEIS A000178).
The first few positive integers that can be written as a product of factorials are 1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, ... (OEIS A001013).
The number of ways that is a product of smaller factorials, each greater than 1, for , 2, ... is given by 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, ... (OEIS A034876), and the numbers of products of factorials not exceeding are 1, 2, 4, 8, 15, 28, 49, 83, ... (OEIS A101976).
The only known factorials which are products of factorials in an arithmetic progression of three or more terms are
(1)
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(2)
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(3)
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(Madachy 1979).
The only solutions to
(4)
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are
(5)
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(6)
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(7)
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(Cucurezeanu and Enkers 1987).
There are no nontrivial identities of the form
(8)
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for with for for except
(9)
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(10)
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(11)
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(12)
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(Madachy 1979; Guy 1994, p. 80). Here, "nontrivial" means that identities with , or equivalently are excluded, since there are many identities of this form, e.g., .
Values of for which can be written as a product of smaller factorials are 1, 4, 6, 8, 9, 10, 12, 16, 24, ... (OEIS A034878).