The Prime Glossary: multifactorial prime

multifactorial prime
(another Prime Pages' Glossary entries)

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It seems natural (to some authors) to generalize the notion of factorial primes by using the multifactorial functions:

n! = (n)(n-1)(n-2)...(1)
n!! = (n)(n-2)(n-4)...(1 or 2)
n!!! = (n)(n-3)(n-6)...(1,2 or 3)

For example, 7! = 5040, 7!! = 105, 7!!!=28, 7!!!! = 21, and 7!!!!! = 14. The double factorial notation is sometimes used in statistics texts for combinatorial arguments.

Since 7!!!!! can be hard to read (those of us getting older lose count of the !'s) and is easy to confuse with the huge number (((((7!)!)!)!)!, we will write 7!₅ (i.e. using 5 as a subscript to the exclamation mark). More generally,

n!_j = n if 0 < n < j, and
n!_j = n ^. (n-j)!_j

Multifactorial primes are primes of the forms n!!+/-1, n!!!+/-1, n!!!!+/-1, and so on.

Ken Davis suggests that we also consider the forms n!!+ 2. Checking to n=5000 he has found:

n!!-2 is prime for n = 5, 7 , 15 , 17, 19, 51, 73, 89, 131, 153, 245, 333, 441, 463, 825, 1771, and 2027.
n!!+2 is prime for n = 3, 5 ,7 ,9, 21, 23, 27, 57, 75, 103, 169, 219, 245, 461, 695, 1169; and probable-prime for n=3597, 3637.

See Also: Factorial, FactorialPrime, PrimorialPrime

Related pages (outside of this work)

Organized searches for (multi)factorial primes (check status or perhaps join in!)
The top twenty primorial, factorial, and multifactorial primes

References:

CD93
C. Caldwell and H. Dubner, "Primorial, factorial and multifactorial primes," Math. Spectrum, 26:1 (1993/4) 1--7.