Wikipedia / Primorial
Wikipedia / Primorial
In mathematics, and more particularly in number theory, primorial", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.
Definition for prime numbers
primes:
is the signifies the product of the first 5 primes:
are:
Asymptotically, primorials grow according to
Definition for natural numbers
In general, for a positive integer is the product of all primes less than or equal to ; that is,
is the prime-counting function . This is equivalent to
represents the product of all primes no greater than :
, this can be calculated as:
Consider the first 12 values of the sequence :
We see that for composite is equal to the preceding term since is composite.
Primorials are related to the first Chebyshev function by
for large values of , primorials therefore grow according to:
Properties
For any iff is the largest prime such that .
be the divisors.
The sum of the reciprocal values of the primorial converges towards a constant
The Engel expansion of this number results in the sequence of the prime numbers. Griffiths (2015) proved that it is irrational.
Euclid's proof of his theorem on the infinitude of primes.
. For , later on.
Since the binomial coefficient and , we have the following upper bound:.
Using elementary methods, Denis Hanson showed that .
. Furthermore, they showed that for , .
Applications
Primorials play a role in the search for prime numbers in additive arithmetic progressions results in a prime, beginning a sequence of thirteen primes found by repeatedly adding , and ending with is also the common difference in arithmetic progressions of fifteen and sixteen primes.
Every highly composite number is a product of primorials.
Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial is smaller than for any positive integer less than is the Euler totient function.
Any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values.
Base systems corresponding to primorials (such as base 30, not to be confused with the primorial number system) have a lower proportion of repeating fractions than any smaller base.
Every primorial is a sparsely totient number.
Compositorial
The -compositorial of a composite number is the product of all composite numbers up to and including . The -compositorial is equal to the -factorial divided by the primorial . The compositorials are
Riemann zeta function
The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function :
.
Table of primorials
See also
Notes
References
Spencer, Adam "Top 100" Number 59 part 4.