A350747 - OEIS

%I #47 Feb 24 2022 19:31:42

%S 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,

%T 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,

%U 0,9,6,6,0,10

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

%C From _Wolfdieter Lang_, Feb 09 2022: (Start)

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

%C 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)

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

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

%e For n = 24, the trajectory is 24, 36, 40, 44, 45, 33, ... so a(24) = 5.

%p with(numtheory): A001222:= n -> bigomega(n):

%p A001414:= proc(n) local e, j; e:=ifactors(n)[2]; add(e[j][1] * e[j][2],j= 1..nops(e)) end proc :

%p B := n-> A001414(n) * A001222(n):

%p 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:

%p 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 :

%p Seq(F(n), n=0..100)

%Y Cf. A349824, A001414, A001222.

%K nonn

%O 0,5

%A _Gary Detlefs_, Jan 13 2022

%E More terms from _Jinyuan Wang_, Jan 15 2022

%E Edited by _Wolfdieter Lang_, Feb 09 2022