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

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, 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, 2, 13, 10, 5, 9, 8, 15, 9, 6, 2, 17, 13, 8, 9, 5, 15, 12

Offset

1,2

Comments

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.

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.

Example with n = 17; start with k = 3.

3^17 = 129140163, sum of the decimal digits = 27,

27^17 = 2153693963075557766310747, sum of the decimal digits = 117,

117^17 = 144264558065210807467328187211661877, sum of the decimal digits = 153,

153^17 = 13796036156758195415808856807283698713, sum of the decimal digits = 189,

189^17 = 501014933601411817143935347829544613629, sum of the decimal digits = 153 is already in the trajectory.

Links

Table of n, a(n) for n=1..80.

Example

0 is in the sequence 1^1 -> 1;
For the power 2, a(2) = 5:
    2 ->       2^2 =   4;
    4 ->       4^2 =  16;
   16 ->   (1+6)^2 =  49;
   49 ->   (4+9)^2 = 169;
  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.

Maple

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:

Crossrefs

Cf. A177148, A182128.

Sequence in context: A090484 A177232 A200099 * A010038 A232109 A270753

Adjacent sequences: A182126 A182127 A182128 * A182130 A182131 A182132

Keyword

nonn,base

Author

Michel Lagneau, Apr 13 2012

Status

approved