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Number of iterations of the map n -> (sum of the decimal digits of n)^3 before reaching the last number of the cycle.
4

%I #16 Jan 30 2022 02:44:25

%S 0,0,2,3,3,2,3,3,1,2,1,2,3,3,2,3,3,1,2,2,2,3,3,2,3,3,1,2,2,2,3,3,2,3,

%T 3,1,2,2,2,2,3,2,3,3,1,2,2,2,2,3,2,3,3,1,2,2,2,2,3,2,3,3,1,2,2,2,2,3,

%U 2,2,3,1,2,2,2,2,3,2,2,3,1,2,2,2,2,3,2

%N Number of iterations of the map n -> (sum of the decimal digits of n)^3 before reaching the last number of the cycle.

%C a(n) is the number of times that the cube of the sum of the digits must be calculated before reaching the last number of the cycle.

%e 0 is in the sequence twice because 0 -> 0 and 1 -> 1;

%e a(3) = 3:

%e 3 -> 3^3 = 27;

%e 27 -> (2+7)^3 = 729;

%e 729 -> (7+2+9)^3 = 18^3 = 5832 is the end of the map because 5832 -> (5+8+3+2)^3 = 18^3 is already in the trajectory. Hence we obtain the map: 3 -> 27 -> 729 -> 5832 with 3 iterations.

%p A182128 := proc(n)

%p local traj ,c;

%p traj := n ;

%p c := [n] ;

%p while true do

%p traj := A118880(traj) ;

%p if member(traj,c) then

%p return nops(c)-1 ;

%p end if;

%p c := [op(c),traj] ;

%p end do:

%p end proc:

%p seq(A182128(n),n=0..80) ; # _R. J. Mathar_, Jul 08 2012

%Y Cf. A177148, A178481, A118880.

%K nonn,base

%O 0,3

%A _Michel Lagneau_, Apr 13 2012