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,](https://mathworld.wolfram.com/images/equations/CarmichaelFunction/NumberedEquation1.gif) |
(1)
|
where 
are primaries.
It can be defined recursively as
 |
(2)
|
Some special values include
![lambda(2^n)=<span style=]() {1 for n=1, n=2; 2 for n=2; 2^(n-2) otheriwse " src="https://mathworld.wolfram.com/images/equations/CarmichaelFunction/NumberedEquation3.gif" style="height:64px; width:188px" /> |
(3)
|
and
![lambda(n!)=<span style=]() {1 for n=1, n=2; 2 for n=3; 4 for n=5; (n!)/(2n#) otherwise, " src="https://mathworld.wolfram.com/images/equations/CarmichaelFunction/NumberedEquation4.gif" style="height:102px; width:199px" /> |
(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 
.
REFERENCES:
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 27, 1989.
Riesel, H. "Carmichael's Function." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 273-275, 1994.
Sloane, N. J. A. Sequences A002322/M0298 and A011773 in "The On-Line Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, p. 226, 1991.
