Fred wries: << A few years ago a local schoolteacher called Neil Hamilton observed that the original Collatz problem, for the function f(x) = 3x+1 if x is odd, x/2 if x is even is equivalent to the analogous problem for the new function g(y) = (y+1)/4 if y mod 4 = 3, (3y+1)/4 if y mod 4 = 1, 3y/2 if y mod 4 = 0,2. While I said at the time that I thought this was probably well-known, I didn't find any mention of it in the literature; and it did occur to me to wonder whether there had been any systematic attempt to investigate such equivalences.

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Yes, this is well-known, and sometimes the way that Jeff Lagarias prefers to define the Collatz function. (My own preference is to let f(N) = (3N+1)/(the largest power of 2 that divides 3N+1). which means the domain and range may be considered to be the odd integers.) --Dan