натуральні числа

The power transformation of Newton's binomial forms two equal 3n±1 algorithms for transformations of numbers n belongs to N, each of which have one infinite cycle with a unit lower limit of oscillations. It is shown that in the reverse direction, the Kollatz sequence is formed by the lower limits of the corresponding cycles, and the last element goes to a multiple of three odd numbers.

It is shown that: 1. The sequence {2⁰,2¹, 2², 2³, 2⁴, 2⁵, 2⁶, 2⁷,2⁸,...} that forms the main graph m=1 of Collatz is related to the power transformation of Newton's binomial (1+1)^ξ, ξ=0, 1, 2, 3,... 2. The main K_main and side m >1 graphs and their corresponding sequences {K_main } and {K_m } are related by the relation {K_m }=m⋅{K_main }. 3.