[seqfan] Three Conjectures

[Tomasz Ordowski]

Dear SeqFans!

Let a(n) be the maximal exponent in prime factorization of n.

The Conjectures:

(*) There are no composite numbers n > 4 such that n == a(n) (mod phi(n)).
By Lehmer's totient conjecture, there are no such squarefree numbers.

(**) There are no odd numbers n such that n == a(n) (mod lambda(n)) with
a(n) > 1.
These are odd numbers n such that a(n) > 1 and b^n == b^a(n) (mod n) for
all b.

(***) There are no odd numbers n such that n == a(n) (mod ord_{n}(2)) with
a(n) > 1.
These are odd numbers n such that a(n) > 1 and 2^n == 2^a(n) (mod n).
Note that (***) ==> (**), so it's not a numerical coincidence.

Are these well known conjectures?
Can anything be proven here?
How far can check them?

Best regards,

Thomas Ordowski
________________
Cf. https://oeis.org/history/view?seq=A051903&v=65
https://oeis.org/A276976 and https://oeis.org/A270096