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Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique realsolution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\geck+4$, we establish an asymptotic lower bound of the expected order ofmagnitude for the number of representations of a large positive integer as thesum of one prime and $s$ positive integral $k$-th powers.

Partitio Numerorum: sums of a prime and a number of $k$-th powers