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Template:Sequence of the Day for July 15

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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
Yesterday's SOTD * Tomorrow's SOTD

The line below marks the end of the <noinclude> ... </noinclude> section.



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  = 1
in
  
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  = 1
in
  i
  
n
i  = 1
¬  (in)
  i
= n!,
where is the co-phi-torial of
n
(product of all positive integers up to and not coprime to
n
), and (or
¬  (in)
) 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)!
.