[seqfan] The ord-radical numbers

[Tomasz Ordowski]

Dear SeqFans!

   Let's define the title numbers:

Numbers n such that ord_{n}(2) < n-1 and rad(ord_{n}(2)) = rad(n-1).

   The prime ord-radical numbers:
17, 41, 97, 109, 113, 137, 193, 251, 257, 281, 307, 313, 353, 401, 409,
433, 449, 521, 569, 577, 593, 617, 739, 761, 769, 809, 811, 857, 929, 977,
...

   The composite ord-radical numbers:
1729, 2431, 6601, 8749, 9605, 10585, 12801, 15211, 30889, 46657, 69751,
88561, 92929, 105001, 196021, 272323, 348161, 368641, 427233, 460801,
468751, 534061, 610051, 622909, 950797, 992251, ...

   The pseudoprime ord-radical numbers:
1729, 6601, 10585, 12801, 30889, 46657, 88561, 196021, 348161, 427233,
534061, 622909, 950797, 1489665, 2628073, 2704801, 3225601, 3316951,
3763801, 6840001, ...

   Theorem:
There are infinitely many the ord-radical numbers.

   Proof:
If n = 2^(2^k)+1, then m = ord_{n}(2) = 2^(k+1),
so m < n-1 for k > 1 and rad(m) = rad(n-1) = 2.

Problems:    Are there infinitely many prime ord-radical numbers?
            Are there infinitely many composite ord-radical numbers?
Hard: Are there infinitely many pseudoprime ord-radical numbers?

   Let us define:
Pseudoprimes n such that (n-1)/ord_{n}(2) = 2^k for some k.
Data: 12801, 348161, 3225601, 104988673, ...
Question: Is this a subset of the ord-radical numbers?
Cf. A243050. Are these the same numbers?
See https://oeis.org/A243050.

Best regards,

Ami Eldar & Tom Ordowski
________________________
Denote: rad(n) = A007947(n),
see https://oeis.org/A007947;
ord_{n}(2) = A002326((n-1)/2),
see https://oeis.org/A002326.