[seqfan] Theorem -- Extension to negative bases

[Tomasz Ordowski]

Dear SeqFans!

Let me remind you of the proven generalization:

Let B be the set of natural numbers n such that

      (b^n-1)/(b-1) == 1 (mod n)

with a fixed integer b > 1.

Then, the number (b^k-1)/(b-1) belongs to the set B
if and only if k belongs to B.

The questions: How to extend this theorem to the integer bases b < -1 ?
It should work with every integer b <> 1.  How will it be for b = 0 and -1
?
Is it necessary to assume that n is odd for b < -1 ?

Best regards,

Thomas

P.S. A comment:
  http://oeis.org/A015919
If 2^k-1 is a term, then so is k.
Asymptotic formula: a(n) ~ n log n.
I still have a full account on the OEIS.



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