primorial
التعريفات والمعاني
== English ==
=== Etymology ===
Blend of prime + factorial. Coined by American engineer and mathematician Harvey Dubner.
=== Pronunciation ===
IPA(key): /pɹaɪ̯ˈmɔːɹi.əl/
Rhymes: -ɔːɹiəl
=== Noun ===
primorial (plural primorials)
(number theory) Any number belonging to the integer sequence whose nth element is the product of the first n primes.
Synonym: primorial number
(number theory) A unary operation, denoted by the postfix symbol # and defined on the nonnegative integers, which maps 0 to 1, 1 to 1, and each subsequent number to the product of all primes less than or equal to it; the value mapped to by said operation for a given input.
2020, Rong Pan, Qinheping Hu, Rishabh Singh, Loris D'Antoni, Solving Problem Sketches with Large Integer Values, Peter Müller (editor), Programming Languages and Systems: 29th European Symposium, Proceedings, Springer, LNCS 12075, page 587,
The following number theory result relates the primorial to the Chebyshev function.
ϑ
(
n
)
=
log
(
n
#
)
=
log
2
(
1
+
o
(
n
)
)
n
=
(
1
+
o
(
n
)
)
n
{\displaystyle \vartheta (n)=\log(n\#)=\log {2^{(1+o(n))n}}=(1+o(n))n}
==== Usage notes ====
The primorial operation may be defined as:
n
#
=
∏
p
≤
n
p
p
r
i
m
e
p
=
∏
i
=
1
π
(
n
)
p
i
=
p
π
(
n
)
#
{\displaystyle n\#=\prod _{p\leq n \atop p~{\mathsf {prime}}}p=\prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#}
,
where
π
(
n
)
{\displaystyle \pi (n)}
denotes the prime-counting function, which gives the number of primes
≤
n
{\displaystyle \leq n}
.
It can also be defined recursively:
n
#
=
{
1
i
f
n
=
0
,
1
(
n
−
1
)
#
×
n
i
f
n
i
s
p
r
i
m
e
(
n
−
1
)
#
i
f
n
i
s
c
o
m
p
o
s
i
t
e
.
{\displaystyle n\#={\begin{cases}1&{\mathsf {if}}~n=0,\ 1\\(n-1)\#\times n&{\mathsf {if}}~n~{\mathsf {is~prime}}\\(n-1)\#&{\mathsf {if}}~n~{\mathsf {is~composite}}.\end{cases}}}
==== Related terms ====
primorial prime
==== Translations ====
=== Further reading ===
Sequence A002110 of the On-Line Encyclopedia of Integer Sequences
Chebyshev function on Wikipedia.Wikipedia (Particularly, first Chebyshev function.)
Sparsely totient number on Wikipedia.Wikipedia