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

[Max A.]

Ralf,

It looks like these conjectures are incorrectly stated in your paper.
Conjectures 111 and 112 both refer to A036556 whose current definition
is simply wrong (and your paper repeats it in a formal form):

%S A036556 7,14,23,27,28,29,31,39,46,54,56,57,58,62,71,78,87,91,92,93,95,103,107,
%N A036556 Multiples of 3 with an odd number of one bits in base 2.

e.g.: the first term 7 is not a multiple of 3, contradicting to the %N field.
What is the correct definition of A036556?


Conjecture 113 seems to be given in a wrong direction in your paper.
You ask to prove that if a(3k)=0 then k belongs to A006288. But it is
opposite to proving that a(3*A006288) = 0.

Max

On 1/4/07, Ralf Stephan <ralf at ark.in-berlin.de> wrote:
> Max, short answer first.
> > 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?
>
> These two refer to the following OEIS entry
>
> %N A065359 Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n.
> %C A065359 Conjectures: a(n) = 3 or -3 iff n in 3*A036556; a(n) = 0 iff n=3k with k not in A036556. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 07 2003
>
>
> > 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?
>
> This refers to:
>
> %C A083905 Conjecture: a(3*A006288) = 0.
>
>
> ralf
>
>