Conjectures 111-113. Smaller counter-example for the second part.

[Antti Karttunen]

Antti Karttunen wrote:

>
> And for the latter part of the conjecture:
> "a(n) = 0 <=> n=3k with k not in A036556."
>
> The ==> direction holds, as a(n) is 0 only for multiples of 3,
> but for the odious multiples of 3 the A065359 must give a non-zero 
> integer.
>
> For <== direction, n must be 3k for some k, that a(n) were zero.
> Is there k, such that 3k is one of the evil numbers, A001969
> for which A065359(3k) is not zero?
> At least 2839893 (1010110101010101010101 in binary) =  946631 * 3
> is, thus giving a counter-example for <== direction of the conjecture.
> (A065359(2839893) gives 9-3 = 6, while there are 9+3 = 12 1-bits).


5463 (1010101010111 in binary) = 3 * 1821
gives a much smaller counter-example, probably the smallest.

BTW, did you Ralph check your conjectures experimentally any further
than how many terms there are in the corresponding A-sequences?


Cheers,

Antti