Two days ago I wrote:

If it *has* been proven that there's no one-sided counterexample, you could still do the above, but then look the the differences between each pair of sums, regarding one sum as representing the +1 digits and the other sum as representing the -1 digits. (Any collisions would be regarded as 0 digits. For instance sum (2/3)^squares minus sum (2/3)^cubes would have (2/3)^sixth-powers in both sequences.) This would be a lot harder, as there would be about 20 billion such differences.

I realized today that this was stupid of me. After generating the 230621 sums (sum (2/3)^j for j in each OEIS entry), which I did and reported on yesterday, I don't have to try all 230621*230620/2 differences to look for a difference of plus or minus one. I can sort the sums, then simultaneously move down the list from the top and up the list from the bottom, and that will quickly find all differences whose absolute values are close to one. I've done so. The only interesting thing I found was that the difference between the sums for A087156 and for A073536 equaled one to fourteen decimal digits. The former is simply all positive integers except one. (2/3)^A087156 sums to 4/3, or would if the terms went on forever. The latter is "Breaking indices for A058842 (i.e. n such that A058842(n) is not equal to 3*A058842 (n-1) )." and A058842 is "From Renyi's "beta expansion of 1 in base 3/2": sequence gives a(1), a(2), ... where x(n) = a(n)/2^n, with 0 < a(n) < 2^n, a(1) = 1, a(n) = 3*a(n-1) modulo 2^n." So, as with A077468, this series is explicitly about base 3/2. (2/3)^A073536 appears to sum to 1/3. The reason the difference differs from one by a few parts per quadrillion is no doubt because the terms in A087156 (intended to be all integers greater than one) stop at 77. So there don't appear to be any sequences in OEIS with a zero asymptotic density of non-zero terms for which (2/3)^j sums to one. Of course there's always more I could try. For instance I could treat every odd term or every odd-placed term as negative and see what happens.

It's certain that many of these 20 billion differences would be very close to 1 just by chance.

More stupidity on my part. With a mere twenty billion differences, whose absolute values are pretty evenly distributed from zero to two, there should be nothing that's much closer to one than one part in twenty billion by mere chance. And indeed that's just what I see.