TOPICS
Search

Primorial


Let p_n be the nth prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by

 p_n#=product_(k=1)^np_k.
(1)

The values of p_n# for n=1, 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).

It is sometimes convenient to define the primorial n# 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 n, i.e.,

 n#=product_(k=1)^(pi(n))p_k,
(2)

where pi(n) is the prime counting function. For n=1, 2, ..., the first few values of n# are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).

PrimorialLimit

The logarithm of p_n# is closely related to the Chebyshev function theta(x), and a trivial rearrangement of the limit

 lim_(x->infty)x/(theta(x))=1
(3)

gives

 lim_(n->infty)(p_n#)^(1/p_n)=e
(4)

(Ruiz 1997; Finch 2003, p. 14; Pruitt), where e is the usual base of the natural logarithm.


See also

Chebyshev Functions, Euclid Number, Factorial, Factorial Prime, Fibonorial, Fortunate Prime, Prime Products, Primorial Prime, Prime Sums, Smarandache Near-to-Primorial Function, Twin Peaks

Explore with Wolfram|Alpha

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." 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."

Referenced on Wolfram|Alpha

Primorial

Cite this as:

Weisstein, Eric W. "Primorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Primorial.html

Subject classifications