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

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

Author

Michel Lagneau, May 28 2010

Keywords

• nonn
• base

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]

Examples

			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.
		

Crossrefs

Cf. A177148, A182128.

Programs

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