[seqfan] Set A121707 and its subsets

[Tomasz Ordowski]

Dear readers!

First, note that n^{n-1} == 1 (mod (n-1)^2) for every n > 1.
Second, notice that n^{n-1} == 1 (mod D_{n-1}) if and only if (n, D_{n-1})
= 1, *
where D_{n-1} is the denominator of Bernoulli number B_{n-1}.
For odd n > 1, D_{n-1} = Product_{p prime, p-1 | n-1} p.
Let's define the subsets of "anti-Carmichael" numbers:
{A} Odd numbers n > 1 such that n^{n-1} == 1 (mod D_{n-1}^2).
{B} Odd numbers n > 1 such that n^{n-1} == 1 (mod (n-1)^2 D_{n-1}).
 Are there odd numbers n > 1 such that n^{n-1} == 1 (mod (n-1)^2 D_{n-1}^2)
?
Odd numbers n > 1 such that n^{n-1} == 1 (mod (n-1) D_{n-1}) are all
A121707.

Best regards,

Thomas Ordowski
________________________
(*) Odd numbers n > 1 such that (n, D_{n-1}) = 1
are those "anti-Carmichael" numbers A121707:
A121707 - OEIS <https://oeis.org/A121707> (see my comments there).