Evil numbers

[Dan Dima]

A comment to A001969 is: "Gelfond conjectured that this sequence has
asymptotic density 1/2."
Is this one still unproved? I searched on the Internet but I did not found
any reference that such a proof exists.

Clearly, 2n+1 is in the sequence iff 2n is not.
But this proves only the fact that the sequence contains the same number of
"evil even" numbers with the number of "odious(not evil) odd" numbers. So if
you argue only this as a proof I think it is clearly wrong!
However this is the first brick of my easy proof...

Consider p as the probability that an integer is evil.
Any even number as binary is ...xxx0 meanwhile for an odd we have
...xxx1 where ...xxx covers all the binaries, too.
So the probability for an integer to be an "evil even" number is 1/2 p and
for an "evil odd" number is 1/2 (1-p), hence:
1/2 p + 1/2 (1-p) = p, then we conclude p = 1/2.

In fact my first proof it was to split all the integers like:
...0 -> 1/2                   n evil iff n+1 not evil
...01 -> 1/4                 n evil iff n+1 evil too
...011 -> 1/8               n evil iff n+1 not evil
...0111 -> 1/16           n evil iff n+1 evil too
......
1/2 (1-p) + 1/4 p + 1/8 (1-p) + 1/16 p + ... = p
(1/2 + 1/8 + 1/32 + ...)(1-2p) = 0 then p = 1/2.

Please correct me if I am wrong.

Regards,
Dan





On 8/23/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> Sequence A001969 (the "evil" numbers, those with an even number of 1's
> in their binary representation) has a comment: "Gelfond conjectured
> that
> this sequence has asymptotic density 1/2."  I'm sure Gelfond was smart
> enough to notice that 2n+1 is in the sequence iff 2n is not; and to
> realize
> that any sequence which contains exactly 1 of 2n and 2n+1 has
> asymptotic
> density 1/2.
>
> So what should this comment really be?  Did Gelfond make his conjecture
> about some similar sequence?  Did he arrive at this sequence in some
> very
> different way, so that it wasn't clearly this sequence?  Or did he
> actually
> prove it - or just state it as obvious - and somebody misread it as a
> conjecture?
>
> Franklin T. Adams-Watters
>
>
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