login
A178481
Number of steps of the map x -> A055566(x), starting at n, before reaching the end of the cycle.
2
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, 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, 4, 1, 4, 1, 2, 2, 3, 2, 3, 4, 2, 4, 2
OFFSET
0,3
COMMENTS
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).
Example:
6 -> 6^5 = 7776 -> (7+7+7+6)^5 = 27^5.
27^5 = 14348907 -> (1+4+3+4+8+9+0+7)^5 = 36^5.
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.
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.
Example with k = 17; start with 3.
3^17 = 129140163, sum = 27,
27^17 = 2153693963075557766310747, sum = 117,
117^17 = 144264558065210807467328187211661877, sum = 153,
153^17 = 13796036156758195415808856807283698713, sum = 189,
189^17 = 501014933601411817143935347829544613629, sum = 153 is already in the set.
[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]
EXAMPLE
a(0) = 0 and a(1) = 0 because 0 -> 0 and 1 -> 1.
a(15) = 1 because 15^5 = 759375 -> (7+5+9+3+7+5) = 36,
36 ^5 = 60466176 -> (6+0+4+6+6+1+7+6) = 36.
MAPLE
A178481 := proc(n)
local traj , c;
traj := n ;
c := [n] ;
while true do
traj := A055566(traj) ;
if member(traj, c) then
return nops(c)-1 ;
end if;
c := [op(c), traj] ;
end do:
end proc:
seq(A178481(n), n=0..80) ; # R. J. Mathar, Jul 08 2012
CROSSREFS
Sequence in context: A342117 A107488 A114236 * A348726 A377405 A171545
KEYWORD
nonn,base
AUTHOR
Michel Lagneau, May 28 2010
STATUS
approved