Here’s the latest progress on the hyperbinary sequence. We’re trying to figure out the inverse relation of the function : given a particular number , where does it occur in the hyperbinary sequence? That is, what are the values of for which ?
There are infinitely many, but in a previous post I argued why we only need to find occurrences at even positions of the sequence, which we call primary occurrences. I have no idea how easy or hard it is to give a general method for finding all primary occurrences. But some progress has been made:
- Brendan proved by induction that and . These correspond to the numbers that occur right next to 1 in the sequence (we saw earlier that ).
- Brendan also proved that . This is impressive, since this pattern certainly isn’t obvious (at least, it wasn’t to me!).
- Fergal Daly conjectured that the number of primary occurrences of is . denotes the so-called Euler totient function; is defined to be the number of positive integers smaller than which are relatively prime to . An explanation of this fact, if it is true (and it really looks like it might be!) would probably go a long way towards finding a general method for computing !
Can anyone find a proof of Fergal’s conjecture?