Mapping Collatz function iterations for all and any N into one Graph...it is possible? Maybe!

Feb 2020
15
2
Australia
I have devised a graph in Mod 9 to illustrate the deterministic path that all iterations of N follow under the Collatz function of n/2 if even and 3n+1 if n is odd. Significant or not? Let’s discuss…

Collatz Chart.jpg
A010878a(n) = n mod 9.
0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0,
1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0,….towards ∞
  • Convert n to n mod 9
  • Locate corresponding node on graph (NB - 0 has been replaced with 9)
  • If n mod 9 is derived from an odd number follow broken edge to next state (lands on iterated state)
  • If n mod 9 is derived from an even number follow unbroken line to next state (lands on iterated state)
  • Ex. n =13 so 4 mod 9 (uneven). Follow broken edge back to 4 which now represents 4 mod 9 even (40)
  • 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
  • (4) (4) (2) (1) (5)(7)(8) (4)(2) (1)
  • ANY iterated sequence of n converted to n mod 9 will follow exact paths predetermined by graph

 

skipjack

Forum Staff
Dec 2006
21,379
2,410
The image looks as though it needs to be tidied up near 1 and 8.