On a Particular Representation of Integers as Sums of kth Powers

Consider the following way of representation of any positive integer x in the form

x = x_l^k+x^k₂+...+x^k_s,                                                   (1)

where x₁, x₂, ..., x_s are integers given by

(x₁+1)^k > x ≥ x₁^k

(x₂+1)^k > x-x₁^k ≥ x₂^k                                                 (2)

................

.................

the process terminating with

(x_s+1)^k > x-x₁^k-x^k₂-........-x^k_s-1 = x^k_s.

It can easily be seen that there is one and only one way of representation of an integer x in this manner. The number of kth powers required in the representation of x in the above manner is clearly a function of x and k, and hence can be denoted by S_k(x).