I am looking for integers which are a (finite) repeating sequence in both base 2 and 3. That is integers which can be expressed as: $$\left(\sum_{i=1}^{M} 2^{i\cdot k}\right) \cdot s \qquad (s<2^{k})$$ and as:
$$\left(\sum_{j=1}^{N} 3^{j\cdot l}\right) \cdot t \qquad (t<2^{l})$$
In particular, I am looking for possible constraints on M, N, s, t, k and l.
As an extra complication, it would suffice to find such a number which has some kind of "head" in front of the repeating sequences, that is:
$$h\cdot2^{(M+1)\cdot k}+\left(\sum_{i=1}^{M} 2^{i\cdot k}\right) \cdot s = h\cdot3^{(N+1)\cdot k}+\left(\sum_{j=1}^{N} 3^{j\cdot l}\right) \cdot t \qquad (t<2^{l}, s<2^{k})$$
How would I go about finding such numbers, and the constraints on numbers of this form? Is there a way this could be impossible for high enough M (and therefore N)?