Conjectures 111-113 from "100 Conjectures from the OEIS"

[Max A.]

On 12/18/06, Ralf Stephan <ralf at ark.in-berlin.de> wrote:

> http://arXiv.org/abs/math.CO/0409509
> "Prove or Disprove. 100 Conjectures from the OEIS."

Could you please clarify the following:

In Conjecture 111:
Let n=21 (=10101 in binary).
Then a_{21}=3 but 21 does not belong to the set { m | m=3k & k=3i &
e_1(k)=1 mod 2 } (simply because all elements of the set are multiples
of 9 while 21 is not).
Is n=21 a counterexample to Conjecture 111?

In Conjecture 112:
Let n=63 (=111111 in binary).
Then a_{63}=0 and m=n/3=21. But 21 belongs to the set { k | k=3i &
e_1(k)=1 mod 2 }.
Is n=63 a counterexample to Conjecture 112?

In Conjecture 113:
Let k=18. Then a_{3k}=a_{54}=0:
a_{54} = 1-a_{27} = 1+a_{13} = 1-a_6 = a_3 = -a_1 = a_0 = 0.
But in the base-4 the last digit of 18 must be different from -1,0,1.
Is k=18 a counterexample to Conjecture 113?

Max