[seqfan] Two questions

[Tomasz Ordowski]

Hello SeqFans!

Let a(n) be the smallest k > 0 such that 2^(n-1) == 2^k (mod n).
a(n) = 1, 1, 2, 2, 4, 1, 3, 3, 2, 1, 10, 3, 12, 1, 2, 4, 8, 5, 18, ...
If n is odd, then n-1 == a(n) (mod A002326((n-1)/2).
Odd numbers n such that a(n) = n-1 are A001122.
An odd composite n is a Fermat pseudoprime if and only if a(n) =
A002326((n-1)/2).
How to prove that a(n) <= A002326((n-1)/2) for all odd n? Checked up to
99999 by Amiram Eldar.

Let b(n) be the smallest m such that b^(n-1) == b^m (mod n) for all b.
b(n) = 1, 1, 2, 3, 4, 1, 6, 3, 2, 1, 10, 3, 12, 1, 2, 7, 16, 5, 18, ...
Odd numbers n such that a(n) = b(n) are A319009.
Note that n-1 == b(n) (mod A002322(n)) for all n.
For n <> 4, n is a prime if and only if a(n) = n-1.
A composite n is a Carmichael number if and only if b(n) = A002322(n).
Numbers n such that b(n) > A002326(n) are 4, 8, 12, 16, 24, 48, 56, 80,
112, 132, 208, 240, 552, 1064, 1104, 1456, 1892, 2128, 4144, 5852, 12208,
17292, 18544, 21424, 22952, ... [data from Amiram Eldar]. How to
independently characterize these numbers?

Best regards,

Thomas Ordowski
___________________________
Cf. A270096 and A276976 (nice).