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 ✓

Yesterday's SOTD * Tomorrow's SOTD

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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  = 1 i⊥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  = 1 i⊥n   i        n ∏   i  = 1 ¬  (i⊥n)   i   = n!,

where $\textstyle {{\overline {\varphi }}_{_{!}}(n)}$ is the co-phi-torial of

n

(product of all positive integers up to and not coprime to

n

), and $\textstyle {i\not \perp n}$ (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)!

.