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

0, 8, 3, 25, 18, 57, 8, 169, 181, 1, 61, 164, 177, 573, 209, 785, 288, 1121, 347, 517, 549, 2219, 53, 481, 871, 3144, 878, 3336, 777, 2369, 996, 1577, 655, 5109, 936, 3040, 5290, 1698, 652, 1349, 4000, 2781, 4083, 5559, 2769, 7834, 7098, 4686, 3451, 14278, 5998

Offset: 1

Author

Michel Lagneau, Apr 15 2012

Keywords

• nonn
• base

Comments

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.

Generalization and conjecture:

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.

Examples

			a(7) = 8 because:
7^7 = 823543;
8^7+2^7+3^7+5^7+4^7+3^7 = 2196163;
2^7+1^7+9^7+6^7+1^7+6^7+3^7 = 5345158;
5^7+3^7+4^7+5^7+1^7+5^7+8^7 = 2350099;
2^7+3^7+5^7+0^7+0^7+9^7+9^7 =  9646378;
9^7+6^7+4^7+6^7+3^7+7^7+8^7 = 8282107;
8^7+2^7+8^7+2^7+1^7+0^7+7^7 = 5018104;
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.
		

Crossrefs

Cf. A182111, A152077, A160862.

Programs

Maple

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: