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Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.
2

%I #19 Sep 16 2017 00:32:42

%S 0,0,5,3,4,4,2,3,2,2,1,3,2,4,2,1,2,1,2,2,5,3,3,1,1,3,1,1,0,1,3,1,2,1,

%T 1,0,0,1,3,1,3,2,2,2,1,1,0,3,2,3,4,2,4,2,1,2,3,1,5,4,2,4,1,2,2,3,1,4,

%U 4,1,4,1,2,2,3,2,3,4,2,4,2

%N Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.

%C a(n) is the number of times taking the 5th powers of the sums of digits before reaching a sum seen before (reaching the last number of the cycle).

%C Example:

%C 6 -> 6^5 = 7776 -> (7+7+7+6)^5 = 27^5.

%C 27^5 = 14348907 -> (1+4+3+4+8+9+0+7)^5 = 36^5.

%C 36^5 = 60466176, last number of the cycle because (6+0+4+6+6+1+7+6)^5 = 36^5 = 60466176 belongs to the list.

%C Generalization for the k-th powers and conjecture: For each k >= 1, iteration of taking the k-th powers of digit sums reaches a cycle.

%C Example with k = 17; start with 3.

%C 3^17 = 129140163, sum = 27,

%C 27^17 = 2153693963075557766310747, sum = 117,

%C 117^17 = 144264558065210807467328187211661877, sum = 153,

%C 153^17 = 13796036156758195415808856807283698713, sum = 189,

%C 189^17 = 501014933601411817143935347829544613629, sum = 153 is already in the set.

%C [It remains unclear whether the author wanted to define iterations of (sumofdigits of n)^5, compatible with A177148 and A182128, or sumofdigits(n^5) here. I've taken the latter to be more compliant with the first terms of the original submission. - _R. J. Mathar_, Jul 08 2012]

%e a(0) = 0 and a(1) = 0 because 0 -> 0 and 1 -> 1.

%e a(15) = 1 because 15^5 = 759375 -> (7+5+9+3+7+5) = 36,

%e 36 ^5 = 60466176 -> (6+0+4+6+6+1+7+6) = 36.

%p A178481 := proc(n)

%p local traj ,c;

%p traj := n ;

%p c := [n] ;

%p while true do

%p traj := A055566(traj) ;

%p if member(traj,c) then

%p return nops(c)-1 ;

%p end if;

%p c := [op(c),traj] ;

%p end do:

%p end proc:

%p seq(A178481(n),n=0..80) ; # _R. J. Mathar_, Jul 08 2012

%Y Cf. A177148, A182128.

%K nonn,base

%O 0,3

%A _Michel Lagneau_, May 28 2010