Has it been proven that there's no *one-sided* counterexample to the Moore-Schulman base-3/2 conjecture? In other words, sum (2/3)^j = 1 for j in some unending integer sequence with an asymptotic density of zero such as the squares, primes, fibonaccis, or factorials? (It's trivial that there's no such *terminating* sequence, for either a one-sides or two-sided counterexample. Any finite sum (or difference) of (2/3)^j for positive integers j would be an even number over an odd number.) If not, has anyone tried summing (2/3)^j for every appropriate sequence in OEIS? (By "appropriate" I mean all terms are positive integers, they increase monotonically, there are infinitely many terms, and the asymptotic density of terms is zero. (A small number of duplicated terms at the beginning of a sequence, as in the factorials and fibonaccis, can be "forgiven.")) If the result is suspiciously close to 1, you could then compute more terms of the series and see if it gets closer and closer to 1. Similarly if the result is very close to some integer power of 2/3, since that's equivalent to the sum over the series with an offset (i.e. adding N to each term). If it does, you could then try to find a proof or disproof that this is not just a coincidence. If there's no such proof and if nobody has tried this, perhaps I will. There are only about 200,000 OEIS entries. And I see that there's an easy way to download all of them, and no rule against doing so for non-commercial purposes. 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. Far more if you allow different offsets for the plus and the minus sequences. It's certain that many of these 20 billion differences would be very close to 1 just by chance. But it might be worth doing with just the most popular OEIS sequences, i.e. those that occur most often in math, in the same sense as pi and e are more popular than the twin primes constant or Khinchin's constant. This popularity probably correlates pretty closely with the more ordinary sort of popularity -- web hit count. Am I correct that the most popular OEIS sequences in both senses of the word are the lowest-numbered ones? I think there's also the possibility that the Moore-Schulman base-3/2 conjecture is false, but that no counteraxmple can ever be found because it would require infinite computation, i.e. more and more backtracking without limit. Any thoughts on this dismal possibility?