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

[Tanya Khovanova]

The correct definition of A036556 seems to be
%N A036556 Numbers m such that 3m has an odd number of one bits in base 2.
or following Tanya's suggestion:
%N A036556 Numbers m such that 3m is odious.

If so, Conjectures 111 and 112 must be corrected as follows:

In Conjecture 111 the line
"\eq n\in\left\{m\,\big|\;m=3k\,\wedge\,k=3i\,\wedge\,e_1(k)\equiv1\bmod2\right\}."
should be replaced with
"\eq n\in\left\{k\,\big|\;k=3i\,\wedge\,e_1(k)\equiv1\bmod2\right\}."

In Conjecture 112 the line
"\eq n=3m\;\wedge\;m\not\in\left\{k\,\big|\;k=3i\,\wedge\,e_1(k)\equiv1\bmod2\right\}."
should be replaced with
"\eq n\in\left\{k\,\big|\;k=3i\,\wedge\,e_1(k)\equiv0\bmod2\right\}."

Max

On 1/5/07, Max A. <maxale at gmail.com> wrote:
> 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
> >
> >
>