A growth model based on the arithmetic Z-game

We present an evolutionary self-governing model based on the numerical atomic rule Z(a,b)=ab/gcd(a,b)2, for a, b positive integers. Starting with a sequence of numbers, the initial generation Gin, a new sequence is obtained by applying the Z-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix TGin is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If Gin is the sequence of positive integers, in the associated matrix remarkable are the distinguished geometrical figures called Z-solitons and the sinuous evolution of the size of numbers on the western edge. We observe that TN* is close to the analogue free of Z-solitons matrix generated from an initial generation in which each natural number is replaced by its largest divisor that is a product of distinct primes. We describe the shape and the properties of this new matrix. N. J. A. Sloane raised a few interesting problems regarding the western edge of the matrix TN*. We solve one of them and present arguments for a precise conjecture on another.