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Two arithmetic inequalities
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Let ƥ(n), σ(n) resp. d(n) denote the classical arithmetic functions representing Euler's totient, the sum of divisors, resp. the number of divisors of n: In this note we offer improvements of the Bagchi{Gupta inequality σ(n) ≥ ƥ(n) + d(n); resp the Makowski inequality nd(n) ≥ƥ(n) + σ(n):
Abstract
Let ƥ(n), σ(n) resp. d(n) denote the classical arithmetic functions representing Euler's totient, the sum of divisors, resp. the number of divisors of n: In this note we offer improvements of the Bagchi{Gupta inequality σ(n) ≥ ƥ(n) + d(n); resp the Makowski inequality nd(n) ≥ƥ(n) + σ(n):
참고문헌 (7)
- H. D. Bagchi, "Problem 343", Jber Deut. Math. Verein, 57, pp.8~9 (1954)
- R. L. Duncan, "Some estimates for σ(n)", Amer. Math. Monthly, 74, pp.713~715 (1967)
- G. H. Hardy, "An introduction to the theory of numbers,4th ed" (1965), Oxford
- A. Makowski, "Problem 339", Elem. Math., 15, pp.39~40 (1960)
- J. Sándor, "Some diophantine equations for particular arithmetic functions (Romanian)", pp.1~10 (1989), Univ. of Timişoara
- J. Sándor, "Handbook of number theory Ⅱ" (2004), Springer Verlag
- J. Sándor, "Handbook of number theory Ⅰ" (2006), Springer Verlag
관련 논문AI추천
ON AN ARITHMETIC INEQUALITY BY K. T. ATANASSOVJozsef Sandor, Lehel Kovacs장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2010 · 제13권 제3호On the Euler minimum and maximum functionsJozsef Sandor장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2010 · 제13권 제2호ON COMPOSITE n DIVIDING '(n)(n) + 1C. Goverdhan, Hussain Abdulkader Al-Aidroos, V.Siva Rama Prasad장전수학회 · Advanced Studies in Contemporary Mathematics2013 · 제23권 제1호On Lehmer’s totient problem and its Unitary analogueC. Goverdhan, Hussain Abdulkader Al-Aidroos, V. Siva Rama Prasad장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2010 · 제13권 제2호On an inequality of KlamkinJózsef Sándor장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2010 · 제13권 제1호Generalizations of number-theoretic sumsNarakorn Rompurk Kanasri, Patchara Pornsurat, Yanapat Tongron대한수학회 · 대한수학회논문집2019 · 제34권 제4호
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