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 Generalizations of number-theoretic sums Commun. Korean Math. Soc. 2019 Vol. 34, No. 4, 1105-1115 https://doi.org/10.4134/CKMS.c180414Published online October 31, 2019 Narakorn Rompurk Kanasri, Patchara Pornsurat, Yanapat Tongron Khon Kaen University; Khon Kaen University; Khon Kaen University Abstract : For positive integers $n$ and $k$, let $S_k(n)$ and $S'_k(n)$ be the sums of the elements in the finite sets $\left\{x^k : 1\leq x\leq n, (x, n)=1\right\}$ and $\left\{x^k : 1\leq x\leq n/2, (x, n)=1\right\}$, respectively. The formulae for both $S_k(n)$ and $S'_k(n)$ are established. The explicit formulae when $k=1, 2, 3$ are also given. Keywords : arithmetic function, Euler's phi-function, M\"obius function, M\"obius inversion formula, sum of power of integers MSC numbers : 11A25 Downloads: Full-text PDF   Full-text HTML