Recursive sufficiency for the Collatz conjecture and computational verification

We define the notion of recursive sufficiency for the Collatz conjecture and we use it to present some results concerning the computational verification of the conjecture. For any integer N ≥ 1 and any recursively sufficient set F, it is proved that all integers in the interval [1, N] satisfy the conjecture if and only if F ∩ [1, N] satisfies the conjecture. We offer a sequence of sieves for which the corresponding sequence of elimination percentages tends to 100\%, and as a result, for any integer P arbitrarily close to 100, we give a sieve whose elimination percentage is at least P%. Also, we prove that if N=2(3^n)+1 is the largest known integer for which all integers 1, 2, ..., N satisfy the conjecture, then all integers N+1, N+2, ..., 2N will satisfy the conjecture as well, and hence, they can be eliminated from the verification process.