Geodesic
The generalized Waring problem: A~new property of positive integers
Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 372-378.

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The paper deals with the problem of whether a positive integer $n>1$ can be written as the sum of $s$ summands that are $r$th powers of integer $s\ge m$, where $m\ge0$ is a chosen integer (for $m=0$ we have the classical Waring problem). For this problem, we define in a natural way arithmetic functions $G(m,r)$ and $g(m,r)$ that are the analogs of the Hilbert functions $G(r)$ and $g(r)$ for the classical Waring problem. It is proved that every positive integer $n$ exceeding some threshold value can be written as the above sum, simultaneously for all $s$, $1\le s\le n$, with a finite number of exceptions, which are determined explicitly. 
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     title = {The generalized {Waring} problem: {A~new} property of positive integers},
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A. A. Zenkin. The generalized Waring problem: A~new property of positive integers. Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 372-378. http://geodesic.mathdoc.fr/item/MZM_1995_58_3_a5/

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