[seqfan] Re: A000108(n) ≡ 1 (mod 6)

L. Edson Jeffery lejeffery2 at gmail.com
Thu Dec 3 04:48:39 CET 2015


Robert,

Amazing. Great work.

After the conjecture I wrote:

"If Conjecture 1 is true, then it should be enough to prove, for all m>8,
that the base 3 representations of both 2^m - 1 and 2^m contain at least
one 2, from which William Keith's conjecture that C(2^m-1) == 0 (mod 3)
would then follow."

At the time, I did not realize the difficulty of the problem. The comment
in A260683 says that Paul Erdős conjectured that there is at least one 2 in
the base 3 representation of 2^n, for all n>8; so, evidently, no proof is
known as of yet. A proof for the case 2^n - 1 is likely no easier.

By the way, the number of twos in the base 3 representation of 2n - 1 was
not in OEIS, so I just proposed it as A265157 (
https://oeis.org/draft/A265157).

Ed Jeffery


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