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

[Ralf Stephan]

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




Zak,  that paper has been scanned in by me and you
can download it from my home page:  se Publ. list / item 117
Neil