논문KCI 등재
Generalizations of number-theoretic sums
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초록
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.
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.
참고문헌 (7)
- John D. Baum, "A Number-Theoretic Sum", Mathematics Magazine, 55(2), pp.111~ (1982), Informa UK Limited[DOI]
- D. M. Burton, "Elementary Number Theory" (2011), McGraw-Hill
- George Mackiw, "A Combinatorial Approach to Sums of Integer Powers", Mathematics Magazine, 73(1), pp.44~46 (2018), Informa UK Limited[DOI]
- P. J. McCarthy, "Introduction to Arithmetical Functions" (1986), Universitext, Springer-Verlag
- I. Niven, "An Introduction to the Theory of Numbers" (1991), John Wiley & Sons
- Thomas J. Pfaff, "Deriving a formula for sums of powers of integers", Pi Mu Epsilon Journal, 12(7), pp.425~430 (2007)
- H. S. Wilf, "Mathematics for the Physical Sciences" (1978), Dover Publications
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