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.
See alsoChebyshev 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 ReferencesFinch, 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|AlphaPrimorial Cite this as:Weisstein, Eric W. "Primorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Primorial.html
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