Wolfram Web Resources » 13,634 entries Last updated: Mon Oct 30 2017 Created, developed, and nurtured by Eric Weisstein at Wolfram Research | Calculus and Analysis > Special Functions > Factorials > Number Theory > Prime Numbers > Prime Sums and Products > Algebra > Products > Interactive Entries > Interactive Demonstrations > # 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. 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 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. http://mathworld.wolfram.com/Primorial.html # Wolfram Web Resources
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