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The 3x + 1 function T(x) takes the values (3x + 1)/2 if x is odd and x/2 if x is even. Let a be any integer with a ¬= 0 (mod 3). If n k (a) counts the number of n with T (k) (n) = a, then for all sufficiently large k, (1.302) k ≤ n k (a) ≤ (1.359) k . If π a (x) counts the number of n with |n| ≤ x which eventually reach a under iteration by T, then for sufficiently large x, π a (x) ≥ x .65 . The proofs are computer-intensive

Density Bounds for the 3x + 1 Problem. I. Tree-Search Method