Conjecture (David E Wilson): The tileable numbers are precisely those n of the form PROD 2^(d_i*b_i)-1)/(2^b_i-1) where d_i >= 1, b_i >= 2, and d_i*b_i | d_(i+1). Proof: We must first define a few terms: Each number corresponds to a binary word formed from its binary digits in order. Whenever we generate terms of this sequence, we will begin with a base word w and an auxiliary word which I will refer to as an "atom". Initially, both consist of the single letter "1". Words corresponding to members of this sequence are formed through two procedures which are iteratively applied to "1": Extension: The atom is appended to w, and Duplication: A run of zeroes with length equal to that of the atom is appended to w, with the constraint that whenever we switch from extending to duplicating and vice versa, the atom is set to the current word w. To illustrate why this is, it is helpful to consider an example: We know that a word for a given term starts with "1". If the next letter is also 1, then we must Extend, and we can continue this process until the next letter is 0. If the next letter is 0, then the space must be filled by an extra copy of w, and we must Duplicate. After Duplication, our setup now looks like this: w0: 111 ... 1 000 ... 0 ? w1: 111 ... 1 000 ... 0 where the zeroes have been added since W0 and W1 cannot overlap. Notice that our atom is now the full string of 1's at the beginning of w. If the next letter of w is again 0, then we must Duplicate again, repeating until the next letter is 1 (or stopping, if we have already generated w) Otherwise, we must Extend. Since the trailing zeroes in w1 will not be filled by w2, we now must have w0: 111 ... 1 000 ... 0 111 ... 1 000 ... 0 ? w1: 111 ... 1 000 ... 0 111 ... 1 000 ... 0 (Our atom is now "111 ... 1 000 ... 0") We may extend further, until the next letter of w is 0, and the process repeats. The conjecture follows, since whenever we begin a run of Extensions or Duplications, the atom has the same length as w, and at the end of such a run, the length of w is always some multiple of the length of the atom, from which we get the divisibility criterion (d_i*b_i | d_(i+1)). Also, Extensions are essentially repeated binary concatenation, which equates to multiplying by some term of the form 2^nk + 2^(n-1)k + ... + 1, which coincides with the form of the terms in the product, and the other criteria follow from the facts that a) w and the atom are never empty and b) each run of Extensions and Duplications had length at least one, so the length of w at least doubles each run. More detailed illustration, showing the role of the atom: 1 (Atom: 1) Extend once: 1 1 (Atom: 11) Duplicate twice: 11 00 00 11 00 00 11 00 00 (Atom: 110000) Extend once: 110000 110000 1100 001100 00 11 000011 0000 (Atom: 110000110000) Duplicate once: 110000110000 000000000000 1100001100 000000000000 00 11000011 000000000000 0000 110000110000 000000000000 1100001100 00000000000000 11000011 0000000000000000 (Atom: 110000110000000000000000) Extend twice: 110000110000000000000000 110000110000000000000000 110000110000000000000000 1100001100000000000000 001100001100000000000000 001100001100000000000000 00 11000011000000000000 000011000011000000000000 000011000011000000000000 0000 etc...