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On Lehmer’s totient problem and its Unitary analogue
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Let Ø be the Euler’s totient function and Ø* be its unitary analogue. For any integer M ≥ 1, let SM = {n ∈ N :MØ(n) = n − 1} and S*M = {n ∈ N : MØ*(n) = n − 1}. If n ∈ S*M with M ≥ 4, we obtain lower bounds for n and ω(n) ( the number of distinct prime factors of n) which significantly improve those given in [1] and [5] for n in SM and S*M respectively.
Abstract
Let Ø be the Euler’s totient function and Ø* be its unitary analogue. For any integer M ≥ 1, let SM = {n ∈ N :MØ(n) = n − 1} and S*M = {n ∈ N : MØ*(n) = n − 1}. If n ∈ S*M with M ≥ 4, we obtain lower bounds for n and ω(n) ( the number of distinct prime factors of n) which significantly improve those given in [1] and [5] for n in SM and S*M respectively.
참고문헌 (11)
- Aleksander Grytczuk, "On the Lehmer Problem Concerning Euler’s Totient Function", Proc. Japan Acad, 79, pp.136~138 (2003)
- Eckford Cohen, "Arithmetical functions associated with the Unitary divisors of an integer", Math. Zeitschs, 74, pp.66~80 (1960)
- Lehmer, D.H, "On Euler’s totient function", Bull. Amer. Math. Soc, 38, pp.745~751 (1932)
- Lieuwens, E., "Do there exist composite numbersMfor which kΦ(M) = M − 1 holds?", Nieuw Archief Voor Wlskunde, 3(18), pp.165~169 (1970)
- Marta Skonieczna, "Some results on the unitary analogue of the Lehmer problem", J. Inequalities in Pure and Applied Mathematics, 9(2) (2008)
- Peter Hagis Jr., "On the equation MΦ(n) = n − 1", Nieuw Arch. Wiskunde, 4(6), pp.225~231 (1988)
- Robin, G., "Estimation de la fonction de Tchebychef θ sur le k-leme nombre premies et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n", Acta Arith., 42, pp.367~389 (1983)
- Roser, J. B., "The nth prime is greater than nlogn", Proc. Lond. Math. Soc., 45, pp.21~44 (1938)
- Sandor, J, "Handbook of number Theory II" (2004), Kluwer Academic publishers
- Subbarao, M.V., "On a problem concerning the Unitary totient function Φ＊(n)", Not. Amer.Math. Soc., 18, pp.740~ (1971)
- Subbarao, M.V., "Some analogues of a Lehmer problem on the totient function", Rocky Mountain J. Math., 15(2), pp.609~620 (1985)
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