Carmichael Function
There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest
integer 
such that 
for all 
relatively
prime to 
. The multiplicative
order of 
(mod 
) is at most 
(Ribenboim 1989). The first few values
of this function, implemented as CarmichaelLambda[n],
are 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, ... (OEIS A002322).
It is given by the formula
![lambda(n)=LCM[(p_i-1)p_i^(alpha_i-1)]_i,](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/CarmichaelFunction/NumberedEquation1.gif) |
(1)
|
where 
are primaries.
It can be defined recursively as
![lambda(n)={phi(n) for n=p^alpha, with p=2 and alpha<=2, or p>=3; 1/2phi(n) for n=2^alpha and alpha>=3; LCM[lambda(p_i^(alpha_i))]_i for n=product_(i)p_i^(alpha_i).](/National_Library/20161007105358im_/http://mathworld.wolfram.com/images/equations/CarmichaelFunction/NumberedEquation2.gif) |
(2)
|
Some special values include
 |
(3)
|
and
 |
(4)
|
where 
is a primorial
(S. M. Ruiz, pers. comm., Jul. 5, 2009).
The second Carmichael's function 
is given
by the least common multiple (LCM) of all
the factors of the totient
function 
, except that if 
, then 
is a
factor instead of 
. The
values of 
for the first few 
are 1, 1, 2, 2,
4, 2, 6, 4, 6, 4, 10, 2, 12, ... (OEIS A011773).
This function has the special value
 |
(5)
|
for 
an odd prime
and 
.
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