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

[Antti Karttunen]

Max A. 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?
>
Currently it is:

%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,
%T A036556 108,109,111,112,113,114,115,116,117,119,123,124,125,127,135,142,151,
%U A036556 155,156,157,159,167,174,182,184,185,186,190,199,206,214
%N A036556 Multiples of 3 with an odd number of one bits in base 2.

Shouldn't it be: "Integers, which when multiplied by 3 yield an odious number (A000069)".
(Or "intersection of A000069 and A008585 (multiples of 3), divided by 3.") ?

And the first half of Ralph's conjecture in http://www.research.att.com/~njas/sequences/A065359

  a(n) = 3 or -3 iff n in 3*A036556; a(n) = 0 iff n=3k with k not in A036556.

then would mean that A065359(n) = 3 or -3 <==> n is an odious number which is multiple of 3.

I think the first counter-example to <== direction of the conjecture is 
87381 = (3*29127) = 10101010101010101 in binary,
whose alternating sum is 9, although it is an odious multiple of 3.
However, ==> direction holds always, because first, of the "11"-divisibility algorithm
(three is the "eleven" of the binary system), the alternating sum
of 3's multiples is always a multiple of 3,
and secondly, if the alternating sum is 3 or -3,
then the number of 1's in odd (or even) positions
is three more than the number of 1's in even (or respectively: odd)
positions. So the total number of 1's is 2*k + 3, that is, odd.
 QED.


Yours,

Antti Karttunen



>
> 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
>>
>>
>