[seqfan] Theorem and Conjecture

[Tomasz Ordowski]

Dear SeqFans!

Theorem: if n(n-1) | 2^n-2, then m(m-1) | 2^m-2 for m = 2^n-1;
                so the sequence A217468 is infinite [sic].

Conjecture: k-1 | 2^k-2 for k = (2^n-1)^3 if and only if n(n-1) | 2^n-2 for
n > 2.

The proof of this theorem is simple, but how to prove this conjecture?

Best regards,

Thomas Ordowski
____________________
https://oeis.org/A069051
https://oeis.org/A217468
https://oeis.org/history/view?seq=A330382&v=23