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

1,2

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.

Links

Table of n, a(n) for n=1..51.

Example

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.

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:

Crossrefs

Cf. A182111, A152077, A160862.

Sequence in context: A065530 A137481 A228886 * A213838 A248294 A004734

Adjacent sequences: A182157 A182158 A182159 * A182161 A182162 A182163

Keyword

nonn,base

Author

Michel Lagneau, Apr 15 2012

Status

approved