cp's OEIS Frontend

A178481 Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.

%I A178481 #19 Sep 16 2017 00:32:42
%S A178481 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 A178481 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 A178481 4,1,4,1,2,2,3,2,3,4,2,4,2
%N A178481 Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.
%C A178481 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 A178481 Example:
%C A178481 6 -> 6^5 = 7776 -> (7+7+7+6)^5 = 27^5.
%C A178481 27^5 = 14348907 -> (1+4+3+4+8+9+0+7)^5 = 36^5.
%C A178481 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 A178481 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 A178481 Example with k = 17; start with 3.
%C A178481 3^17 = 129140163, sum = 27,
%C A178481 27^17 = 2153693963075557766310747, sum = 117,
%C A178481 117^17 = 144264558065210807467328187211661877, sum = 153,
%C A178481 153^17 = 13796036156758195415808856807283698713, sum = 189,
%C A178481 189^17 = 501014933601411817143935347829544613629, sum = 153 is already in the set.
%C A178481 [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 A178481 a(0) = 0 and a(1) = 0 because 0 -> 0 and 1 -> 1.
%e A178481 a(15) = 1 because 15^5 = 759375 -> (7+5+9+3+7+5) = 36,
%e A178481 36 ^5 = 60466176 -> (6+0+4+6+6+1+7+6) = 36.
%p A178481 A178481 := proc(n)
%p A178481         local traj ,c;
%p A178481         traj := n ;
%p A178481         c := [n] ;
%p A178481         while true do
%p A178481                 traj := A055566(traj) ;
%p A178481                 if member(traj,c) then
%p A178481                         return nops(c)-1 ;
%p A178481                 end if;
%p A178481                 c := [op(c),traj] ;
%p A178481         end do:
%p A178481 end proc:
%p A178481 seq(A178481(n),n=0..80) ; # _R. J. Mathar_, Jul 08 2012
%Y A178481 Cf. A177148, A182128.
%K A178481 nonn,base
%O A178481 0,3
%A A178481 _Michel Lagneau_, May 28 2010