The term "Carmichael function" has two definitions. One is the reduced totient function (also called the least universal exponent function), defined
as the smallest positive integer 
such that 
for all integers 
relatively
prime to the positive integer 
. For 
and an integer 
relatively prime to 
, its multiplicative
order (mod 
)
divides 
(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[lambda(p_i^(alpha_i))]_i,](/images/equations/CarmichaelFunction/NumberedEquation1.svg) |
(1)
|
where 
is the prime factorization of 
, with distinct primes 
and positive
integer exponents 
, so 
are the primaries
of 
.
The initial value is 
.
For 
,
it can be defined recursively as follows, where 
is a prime number and 
is a positive
integer:
![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) with at least two distinct prime factors.](/images/equations/CarmichaelFunction/NumberedEquation2.svg) |
(2)
|
Here 
is the totient function. For positive
integers 
,
some special values include
 |
(3)
|
and
 |
(4)
|
where 
is a primorial (S. M. Ruiz, pers. comm.,
Jul. 5, 2009).
The second convention, denoted 
, takes the least
common multiple (LCM) of the totient function
values of the prime power factors of 
,
![lambda^'(n)=LCM[(p_i-1)p_i^(alpha_i-1)]_i,](/images/equations/CarmichaelFunction/NumberedEquation5.svg) |
(5)
|
with 
.
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
 |
(6)
|
for 
an odd prime and 
a positive integer.
