On natural numbers as sums of consecutive \(h\)-th powers (Q1385266)

For a positive integer \(h\geq 2\), let \({\mathcal T}_h\) denote the set of all sums of consecutive \(h\)'th powers, i.e. of the numbers \(n=\sum^{a+b}_{j=a+1} j^h\) \((a\in \mathbb{N}_0, b\in\mathbb{N})\). \textit{S. Platiel} and \textit{J. Rung} [Exp. Math. 12, 353-361 (1994; Zbl 0818.11008)] showed that \({\mathcal T}_2\) is an additive basis of exact order 3. In the present article the author studies the quantities \(T_h(x) =\#\{n\leq x,n\in {\mathcal T}_h\}\), \(P_h(x)=\# \{p\leq x,p \in {\mathcal T}_h\}\). He derives lower and upper bounds (which differ by a power of \(\log x)\) for \(T_h(x)\), and and upper bound for \(P_h(x)\). The proofs are elementary and make use of Bernoulli's formula for the sum \(\sum^n_{j=1} j^h\).