Intended for: July 15, 2012
Timetable
- First draft entered by Alonso del Arte on April 29, 2011 as a verbatim copy of a write-up from November 4, 2010. ✓
- Draft reviewed by Daniel Forgues on May 6, 2011 ✓, July 14, 2018 ✓
- Draft approved by Peter Luschny on July 14, 2011 ✓
The line below marks the end of the <noinclude> ... </noinclude> section.
A001783:
-
phi-torial, or
phi-torial of
, is the product of all positive integers up to and
coprime to
.
-
Here
means that
and
are relatively prime, and
is the
GCD of
and
.
-
{ 1, 1, 2, 3, 24, 5, 720, 105, 2240, ... }
The phi-torial of
is a
divisor of the
factorial of
, since
-
φ!(n) x̅φ!(n) = i i = n!, |
where
is the co-phi-torial of
(product of all positive integers up to and not coprime to
), and
(or
) means that
and
are not relatively prime. Thus
-
We take the positive integers below
, cull out those
that have
prime factors in common with
and then multiply the residual together.
For example, with
, we cull out
2, 4, 6, 8, and multiply
1, 3, 5, 7, giving
105. Of course when
is
prime this works out to
.