Evil numbers
franktaw at netscape.net
franktaw at netscape.net
Wed Aug 23 21:59:53 CEST 2006
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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