Primorial

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.

REFERENCES:

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.

Pruitt, C. D. "A Theorem & Proof on the Density of Primes Utilizing Primorials." https://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."