3n+1 & n/2 we have multiplicative operation and inverse multiplicative operation. But in the case of odd element we have an extra additive operation involved. This makes any sort of analysis very difficult as it is not know how to combine additive and multiplicative operations together in a series of transformation. If we had known how to solve additive and multiplicative operations together in a series of transformations, primes would have been much easier to deal with. In order to deal with this limitation; We create a function that gives same results that the transformations would yield without really applying them. The function shall be represented as [r_b]. We define the relationships between [r_b] & n and various forms of [r_b] We analyze the conditions for failure of the conjecture and test them using [r_b]