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Numbers similar to those of van der Waerden are examined by considering sequences of positive integers \(\{x_1, x_2, \ldots, x_n\}\) with \(x_{i+1} = x_i + d + r_i\), where \(d \in {Z}^+\) and \(0 \leq r_i \leq \max(0, f(i))\) for a given function \(f\) defined on \({Z}^+\). Let \(w_f(n)\) denote the least positive integer such that if \(\{1, 2, \ldots, w_f(n)\}\) is \(2\)-colored, then there exists a monochromatic sequence of the type just described. Tables are given of \(w_f(n)\) where \(f(i) = i – k\) for various constants \(k\), and also where \(f(i) = i\) if \(i \geq 2\), \(f(1) = 0\). In this latter case, as well as for \(f(i) = i \), an upper bound is given that is very close to the actual values. A tight lower bound and fairly reasonable upper bound are given in the case \(f(i) = i – 1\).