I have been trying to answer the following question: Is there a general formula for Carmichael's totient function ($\lambda(n)$) that is analogous to the Euler product formula for Euler's totient function ($\varphi(n)$) (where $n \neq p^k$/when $n$ is composite)?
What I have done so far:
I know that $\varphi(n) = n (1 -\frac{1}{p_1})(1 -\frac{1}{p_2})\dots(1 -\frac{1}{p_k})$
(where $p_k$ denotes the $k$th prime that is a factor of $n$)
I also know that $\lambda(n)$ is equivalent to finding the smallest integer $m$ such that $a^m \equiv 1 (\text{mod}$ $n)$ holds for every integer $a$ that is coprime to $n$.
However, I can't proceed any further. Any tips or help would be much appreciated.
Edit 1: Although I can now express $\lambda(n)$ in terms of $\varphi(n)$ for $n = p^k$ (which is given in Relationship between the Carmichael function and Euler's totient function), I now want to know if there is a general formula for $\lambda(n)$ in terms of $\varphi(n)$ where $n \neq p^k$ (i.e when $n$ is composite).
Edit 2: I just recently found out about Euler's totient theorem (i.e. $a^{\varphi(n)} \equiv 1 (\text{mod}$ $n)$ ) and recently learnt that $\lambda(n)=\text{lcm}(\lambda(p_1^{r_1}), \lambda(p_2^{r_2}), \cdots, \lambda(p_m^{r_m}))$ but still no breakthrough.
