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In this article, we define a very important sequence of functions, all thefunctions of this sequence present behaviors very close to that of the Collatzfunction. The study of such functions allows us to obtain very interestingresults concerning the collatz conjecture and more precisely concerning thegeneralization of this problem, the nature of the Collatz function, on thestatement of the Collatz conjecture we show that this conjecture can be statedfor each function on a well-determined subset of Z and other interestingproperties in relation to the behaviors of Collatz sequences are studied andproved in this article.

The sequence of Collatz functions, exceptionality of the 3n+1 function and the notion of Collatz generalized Matrix