[seqfan] Sequences A000790 and A090088 in new versions

[Tomasz Ordowski]

Dear number lovers!

Let a(n) be the smallest composite k such that n^k == -n (mod k) for n > 1.
6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, *10*,
6, 4, 4, 6, 6, 4, 4, 6,
*10*, 4, 4, 38, 6, 4, 4, 6, 6, 4, 4, 6, 46, 4, 4, 10, 6, 4, 4, 6, 6, 4, 4,
6, *58*, 4, 4, *62*, 6, 4, 4, ...
Is this sequence periodic (with the same long period 23#277# as the primary
pretenders)?
It seems that a(n) >= A000790(n+1) for n > 1, with mostly equality; a(n) >
A000790(n+1) for
n = 25, 34, 58, 61, 73, 82, 97, 106, 133, 142, 166, 169, 178, 193, 226,
241, 253, 262, 277, ...
Note that for the above numbers A000790(n+1) is odd.  Conjecture: for n >
1;
if A000790(n+1) is even, then a(n) = A000790(n+1), otherwise a(n) >
A000790(n+1).

Let b(n) be the smallest k > 2 such that (2n-1)^(k-1) == -1 (mod k) for n >
1.
b(n) is the smallest even k > 2 such that (2n+1)^(k-1) == 1 (mod k) for n >
1,
except n = 191, 199, 277, 337, 373, 389, 401, 431, 433, 443, 499, 521, ...
i.e. b(n) is mostly the smallest even pseudoprime to base 2n+1;
A090088(n+1) for n > 1 [to base 2n-1].

How can we explain these coincidences between such different congruences?

Thanks to Ami Eldar for the data and verification of my working
conjectures!

Best,

Tom Ordo
____________
https://oeis.org/A000790 (Fermat primary pretenders).
See also https://oeis.org/A309316 (Euler primary pretenders).
https://oeis.org/A090088 (smallest even pseudoprimes to odd base).