CARMICHAEL's FUNCTION
\[\lambda (n)\]
$$\lambda ~~~~ lambda $$
\[\lambda _{Carmichael}(n) \left\{\begin{matrix} =\phi (n) ~if~n~has~a~primitive~root,~terms~of~A033948 \\ \geqslant \frac{\phi (n)}{2}~ otherwise\ if\ n\ is~term~of~A033949 \end{matrix}\right.\]
A002322 Carmichael's function.
1,1,2,2,4,2,6,2,6,4,10,2,12,6,4,4,16,6,18,4,6,10,22,2,20,12
A033948 Numbers that have a primitive root.
1,2,3,4,5,6,7,9,10,11,13,14,17,18,19,22,23,25,26,27,29,31,34,37
A033949 Numbers that don't have a primitive root.
8,12,15,16,20,21,24,28,30,32,33,35,36,39,40,42,44,45,48,51,52,55
A010554 phi(phi(n)) Number of primitive roots.
1,1,1,1,2,1,2,2,2,2,4,2,4,2,4,4,8,2,6,4,4,4,10,4,8,4,6,4,12
table([(n,euler_phi(euler_phi(n))) for n in [1..60]], header_row=['n', 'EulerPhi(EulerPhi(n))'])
A046144 Number of primitive roots modulo n.
1,1,1,1,2,1,2,0,2,2,4,0,4,2,0,0,8,2,6,0,0,4,10,0,8,4,6,0,12
\[\lambda _{Carmichael}(n) = \phi (n) \\~\\~~~if~and~only~if~n~= \left\{\begin{matrix} 1,2,4; \\ 3, 5, 7, 9, 11, 13, 17, 19,...~A061345; \\6, 10, 14, 18, 22, 26,...~2*A061345. \end{matrix}\right.\]
A061345 Odd prime powers.
1,3,5,7,9,11,13,17,19,23,25,27,29,31,37,41,43,47,49,53,59,61,67