Let be the th prime, then the primorial
(which is the analog of the usual factorial for prime
numbers) is defined by
(1)
The values of
for , 2, ..., are 2, 6, 30, 210, 2310,
30030, 510510, ... (OEIS A002110).
It is sometimes convenient to define the primorial for values other than just the primes, in which case it is
taken to be given by the product of all primes less than or equal to , i.e.,
(2)
where is the prime
counting function. For ,
2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS
A034386).
The logarithm of
is closely related to the Chebyshev function , and a trivial rearrangement
of the limit
(3)
gives
(4)
(Ruiz 1997; Finch 2003, p. 14; Pruitt), where e
is the usual base of the natural logarithm.
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Pruitt,
C. D. "A Theorem & Proof on the Density of Primes Utilizing Primorials."
http://www.mathematical.com/mathprimorialproof.html.Ruiz,
S. M. "A Result on Prime Numbers." Math. Gaz.81, 269,
1997.Sloane, N. J. A. Sequence A002110/M1691
and A034386 in "The On-Line Encyclopedia
of Integer Sequences."