Intended for: July 15, 2012
Timetable
 First draft entered by Alonso del Arte on April 29, 2011 as a verbatim copy of a writeup 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 ✓
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A001783:

phitorial, or
phitorial 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 phitorial of
is a
divisor of the
factorial of
, since

φ!(n) x̅φ!(n) = i i = n!, 
where
$\textstyle {{\overline {\varphi }}_{_{!}}(n)}$ is the cophitorial of
(product of all positive integers up to and not coprime to
), and
$\textstyle {i\not \perp n}$ (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
.