Carmichael function

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In the Carmichael function $\scriptstyle m=\lambda (n)\,$ is $\scriptstyle m\,$ the smallest number so that $\scriptstyle b^{m}\equiv 1{\pmod {n}}\,$ for every $\scriptstyle b\,$ that is coprime to $\scriptstyle n\,$. The namesake is Robert Daniel Carmichael (1879—1967).

Calculation of the Carmichael function

${\begin{aligned}\lambda (1)&=1\\\lambda (2)&=1\\\lambda (4)&=2\\\lambda (2^{k})&=2^{k-2}\quad \mathrm {f{\ddot {u}}r} \ k>2\\\lambda (p^{k})&=p^{k-1}\cdot (p-1)\quad \mathrm {f{\ddot {u}}r} \ p\geq 3\ \mathrm {prim} ,k>0\\\lambda (p_{1}^{k_{1}}p_{2}^{k_{2}}\cdot \cdot \cdot p_{s}^{k_{s}})&=\operatorname {lcm} \{\lambda (p_{1}^{k_{1}}),\lambda (p_{2}^{k_{2}}),...,\lambda (p_{s}^{k_{s}})\}\end{aligned}}$

$p_{i}$ means distinct primes and $k_{i}$ means natural numbers.
The function is available in Mathematica as "CarmichaelLambda[n]".

Carmichael function and Euler totient function

The Carmichael function $\scriptstyle \lambda (n)$ is a divisor of Euler's totient function $\scriptstyle \varphi (n)$ . For 1 and every prime number the Carmichael function is equal to the Euler's totient function

Example of difference

• Euler's totient function:

$\varphi (105)=\varphi (3\cdot 5\cdot 7)=\varphi (3)\cdot \varphi (5)\cdot \varphi (7)=2\cdot 4\cdot 6=48$

• Carmichael function:

$\lambda (105)=\lambda (3\cdot 5\cdot 7)=\operatorname {lcm} \{\lambda (3),\lambda (5)\lambda (7)\}=\operatorname {lcm} \{2,4,6\}=12$

Sequences

The Carmichael function (Cf. A002322):

{1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, ...}