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Template:Sequence of the Day for July 15

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

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A001783:
 n
-phi-torial, or phi-torial of
 n
, is the product of all positive integers up to and coprime to
 n
.
φ!(n) :=
 n ∏ i  = 1i⟂n

i =
 n ∏ i  = 1(i, n)  = 1

i, n ≥ 1.
Here
 i ⟂ n
means that
 i
and
 n
are relatively prime, and
 (i, n)
is the GCD of
 i
and
 n
.
{ 1, 1, 2, 3, 24, 5, 720, 105, 2240, ... }
The phi-torial of
 n
is a divisor of the factorial of
 n
, since
φ!(n) x̅φ!(n) =
 n ∏ i  = 1i⟂n
i

 n ∏ i  = 1¬  (i⟂n)
i
= n!,
where ${\displaystyle \textstyle {{\overline {\varphi }}_{_{!}}(n)}}$ is the co-phi-torial of
 n
(product of all positive integers up to and not coprime to
 n
), and ${\displaystyle \textstyle {i\not \perp n}}$ (or
 ¬  (i⟂n)
) means that
 i
and
 n
are not relatively prime. Thus
φ!(n) =
 n! x̅φ!(n)
.
We take the positive integers below
 n
, cull out those
 i
that have prime factors in common with
 n
and then multiply the residual together. For example, with
 n = 8
, we cull out 2, 4, 6, 8, and multiply 1, 3, 5, 7, giving 105. Of course when
 n
is prime this works out to
φ!(n) =
 n! n
= (n  −  1)!
.