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A182129 Number of iterations of the orbit n -> (sum of the decimal digits of n)^n starting with n, needed to stabilize.

%I A182129 #18 Jan 28 2022 07:47:59
%S A182129 0,5,3,3,5,3,3,5,5,1,6,6,2,6,9,5,2,6,2,7,5,5,6,6,6,3,5,2,9,7,6,13,12,
%T A182129 9,5,9,2,10,9,7,15,9,7,4,7,2,6,3,7,12,6,9,9,5,2,10,12,10,14,7,8,8,11,
%U A182129 2,13,10,5,9,8,15,9,6,2,17,13,8,9,5,15,12
%N A182129 Number of iterations of the orbit n -> (sum of the decimal digits of n)^n starting with n, needed to stabilize.
%C A182129 a(n) is the number of times you form the n-power of the sum of the digits before reaching the last number of the cycle.
%C A182129 Generalization and conjecture: Let k be a positive integer. The number of iterations of the orbit k -> (sum of the decimal digits of k)^n is finite for any exponent n and any starting value k.
%C A182129 Example with n = 17; start with k = 3.
%C A182129 3^17 = 129140163, sum of the decimal digits = 27,
%C A182129 27^17 = 2153693963075557766310747, sum of the decimal digits = 117,
%C A182129 117^17 = 144264558065210807467328187211661877, sum of the decimal digits = 153,
%C A182129 153^17 = 13796036156758195415808856807283698713, sum of the decimal digits = 189,
%C A182129 189^17 = 501014933601411817143935347829544613629, sum of the decimal digits = 153 is already in the trajectory.
%e A182129 0 is in the sequence 1^1 -> 1;
%e A182129 For the power 2, a(2) = 5:
%e A182129     2 ->       2^2 =   4;
%e A182129     4 ->       4^2 =  16;
%e A182129    16 ->   (1+6)^2 =  49;
%e A182129    49 ->   (4+9)^2 = 169;
%e A182129   169 -> (1+6+9)^2 = 256 is the end of the cycle because 256 -> (2+5+6)^2 = 169 is already in the trajectory. Hence we obtain the map: 2 -> 4 -> 16 -> 49 -> 169 -> 256 with 5 iterations.
%p A182129 with(numtheory) : T :=array(1..500) :W:=array(1..500):for n from 1 to 80 do : k:=0:nn:=n:for it from 1 to 50 do:T :=convert(nn,base,10) :l:=nops(T):s1:=sum(T[i],i=1..l):s:=s1^n:k:=k+1:W[k]:=s:nn:=s:od: z:= [seq(W[i],i=1..k)]:V:=convert(z,set):n1:=nops(V): printf(`%d, `,n1):od:
%Y A182129 Cf. A177148, A182128.
%K A182129 nonn,base
%O A182129 1,2
%A A182129 _Michel Lagneau_, Apr 13 2012