Carmichael lambda

Carmichael lambda is the smallest positive integer $m$ such that

$a^m = 1(mod n)$ $\forall$ a coprime to n.

Definition

For a number: $n = {p_1}^{r_1}.{p_2}^{r_2}....{p_k}^{r_k}$ Carmichael lambda function is defined as $\lambda(n) = lcm(\lambda({p_1}^{r_1}), \lambda({p_2}^{r_2}),....\lambda({p_k}^{r_k}))$ where $\lambda(p^r)$ is defined as:

$\lambda(p^r) = \begin{cases} \frac{1}{2} \phi(p^r) & \text{ if } p = 2 \text{ and } r \geq 3 \\ \phi(p^r) & \text{otherwise} \end{cases}$

where $\phi$ is Euler's totient function.and it is given by

${\phi(p^r) = p^{r-1}.(p-1)}$