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A182129 Number of iterations of the orbit n -> (sum of the decimal digits of n)^n starting with n, needed to stabilize. 0
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A090484 A177232 A200099 * A010038 A232109 A270753
KEYWORD
nonn,base
AUTHOR
Michel Lagneau, Apr 13 2012
STATUS
approved

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Last modified July 21 15:35 EDT 2024. Contains 374474 sequences. (Running on oeis4.)