A182160 - OEIS

%I #10 Apr 21 2012 14:52:39

%S 0,8,3,25,18,57,8,169,181,1,61,164,177,573,209,785,288,1121,347,517,

%T 549,2219,53,481,871,3144,878,3336,777,2369,996,1577,655,5109,936,

%U 3040,5290,1698,652,1349,4000,2781,4083,5559,2769,7834,7098,4686,3451,14278,5998

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

%C 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 Generalization and conjecture:

%C 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 a(7) = 8 because:

%e 7^7 = 823543;

%e 8^7+2^7+3^7+5^7+4^7+3^7 = 2196163;

%e 2^7+1^7+9^7+6^7+1^7+6^7+3^7 = 5345158;

%e 5^7+3^7+4^7+5^7+1^7+5^7+8^7 = 2350099;

%e 2^7+3^7+5^7+0^7+0^7+9^7+9^7 =  9646378;

%e 9^7+6^7+4^7+6^7+3^7+7^7+8^7 = 8282107;

%e 8^7+2^7+8^7+2^7+1^7+0^7+7^7 = 5018104;

%e 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 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 Cf. A182111,  A152077, A160862.

%K nonn,base

%O 1,2

%A _Michel Lagneau_, Apr 15 2012