A350747 Number of iterations required to terminate trajectory mapping described in A349824.

0, 1, 0, 0, 8, 0, 9, 0, 7, 4, 8, 0, 3, 0, 7, 8, 7, 0, 6, 0, 1, 2, 2, 0, 5, 2, 1, 0, 0, 0, 0, 0, 6, 0, 7, 6, 4, 0, 6, 7, 3, 0, 5, 0, 2, 1, 6, 0, 8, 1, 5, 4, 5, 0, 3, 7, 6, 3, 11, 0, 9, 0, 10, 8, 11, 5, 9, 0, 9, 6, 6, 0, 10

Offset: 0

Author

Gary Detlefs, Jan 13 2022

Keywords

• nonn

Comments

From Wolfdieter Lang, Feb 09 2022: (Start)

Conjecture from A349824: the iteration f: n -> A349824(n) becomes periodic for each n >= 0.

a(n) gives the number of steps from n to reach the first member of the periodic part. There are the two length 2 periods: (33,28) and (28,33). (End)

It appears that the only nonprime values of n for which a(n) = 0 are {0, 27, 28, 30, 33}.

Examples

			For n = 6, the trajectory is 6, 10, 14, 18, 24, 36, 40, 44, 45, 33, ... so a(6) = 9.
For n = 24, the trajectory is 24, 36, 40, 44, 45, 33, ... so a(24) = 5.
		

Crossrefs

Cf. A349824, A001414, A001222.

Programs

Maple

with(numtheory): A001222:= n -> bigomega(n):
A001414:= proc(n) local e, j; e:=ifactors(n)[2]; add(e[j][1] * e[j][2],j= 1..nops(e)) end proc :
B := n-> A001414(n) * A001222(n):
g:= proc(n) if isprime(n) or n=0 or n=27 or n=28 or n=30 or n=33 then return 0 else return 1 fi end proc:
F:= proc(n) local v,i; v:=n;if n = 1 then return 1 else if g(n)=0 then return 0 else for i from 0 to 100 do v:= B(v);if  v=27 or v=28 or v=30 or v=33 then return i+1; i:=100 fi od fi fi end proc :
Seq(F(n), n=0..100)

Extensions

More terms from Jinyuan Wang, Jan 15 2022

Edited by Wolfdieter Lang, Feb 09 2022