So is there a way to quickly fill a state that represents a number \(N\) with the correct prime divisors. That is the problem of factorisation. Well, in effect the statement of the problem is one solution. If one already knows the primes, then similar clock like constructions can be made...
Start the state zero with a clock which is 1 (or infinity here), then let the clocks drop through the states, each clock in each column has a different period. in effect the same thing can be done with floor of |cos| function. In effect the state vector becomes \[|N> = |\bigg\lfloor cos^2 \bigg(\frac{\pi(N-2)}{2}\bigg)\bigg\rfloor,\bigg\lfloor cos^2 \bigg(\frac{\pi(N-3)}{3}\bigg)\bigg\rfloor,...>\]
but this will only fill up in a fermionic form... There is no 2 in the 2 column for N=4 just a 1 signifying that it is divisible.