cp's OEIS Frontend

A182160 Number of iterations of the map n -> sum of the n-powers of the decimal digits of n.

%I A182160 #10 Apr 21 2012 14:52:39
%S A182160 0,8,3,25,18,57,8,169,181,1,61,164,177,573,209,785,288,1121,347,517,
%T A182160 549,2219,53,481,871,3144,878,3336,777,2369,996,1577,655,5109,936,
%U A182160 3040,5290,1698,652,1349,4000,2781,4083,5559,2769,7834,7098,4686,3451,14278,5998
%N A182160 Number of iterations of the map n -> sum of the n-powers of the decimal digits of n.
%C A182160 a(n) is the number of times you form the sum of the n-power of each digit of n before reaching the last number of the cycle.
%C A182160 Generalization and conjecture:
%C A182160 Let a number k. The number of iterations of the orbit k -> sum of the n - power of the decimal digits of k is finite for any exponent n and any starting value k.
%e A182160 a(7) = 8 because:
%e A182160 7^7 = 823543;
%e A182160 8^7+2^7+3^7+5^7+4^7+3^7 = 2196163;
%e A182160 2^7+1^7+9^7+6^7+1^7+6^7+3^7 = 5345158;
%e A182160 5^7+3^7+4^7+5^7+1^7+5^7+8^7 = 2350099;
%e A182160 2^7+3^7+5^7+0^7+0^7+9^7+9^7 =  9646378;
%e A182160 9^7+6^7+4^7+6^7+3^7+7^7+8^7 = 8282107;
%e A182160 8^7+2^7+8^7+2^7+1^7+0^7+7^7 = 5018104;
%e A182160 5^7+0^7+1^7+8^7+1^7+0^7+4^7 = 2191663 is the end of the cycle with 8 iterations because 2191663-> 2^7+1^7+9^7+1^7+6^7+6^7+3^7 = 5345158 is already in the trajectory.
%p A182160 with(numtheory) : T :=array(1..20000) :W:=array(1..20000):for n from 1 to 85 do : k:=0:nn:=n:for it from 1 to 20000  do:T :=convert(nn, base, 10) :l:=nops(T):s:=sum(T[i]^n, i=1..l):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 %d \n",n,n1):od:
%Y A182160 Cf. A182111,  A152077, A160862.
%K A182160 nonn,base
%O A182160 1,2
%A A182160 _Michel Lagneau_, Apr 15 2012