Very interesting. Which leads me to wonder what reciprocal integers look like in factorial base, where x ∊ (0, 1) is represented as x = ∑ a_n / n! where the sum is over n ≥ 2 with a_n ∊ Z+, 0 ≤ a_n < n, using the greedy algorithm. —Dan
On Saturday/30January/2021,
at 2:05 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote: I wondered what reciprocals of integers looked like in binary, so I computed them. I noticed that lots of them had first and second halves of the repeating section that were complements of each other. For instance 1/97 is 0.000000101010001110100000111111010101110001011111.... That's 0.000000101010001110100000 111111010101110001011111....
For which numbers is this the case? It turns out to be precisely those in http://oeis.org/A014657 (except the first two) times any power of two (including zero). And I see why.