Abstract
In this note, we show that the number of composite integers n ≤ x such that ϕ(n)|n − 1 is at most O(x 1/2(log log x)1/2), thus improving earlier results by Pomerance and by Shan.
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References
Lehmer, D. H.: On Euler’s totient function. Bull. Amer. Math. Soc., 38, 745–757 (1932)
Pomerance, C.: On composite n for which ϕ(n)|n − 1, II. Pacific J. Math., 69, 177–186 (1977)
Shan, Z.: On composite n for which ϕ(n)|n − 1. J. China Univ. Sci. Tech., 15, 109–112 (1985)
Tenenbaum, G.: Introduction to analytic and probabilistic number theory, Cambridge Univ. Press, Cambridge, 1985
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This work is done during a visit by the second author to the University of Missouri, Columbia; the hospitality and support of this institution are gratefully acknowledged
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Banks, W.D., Luca, F. Composite Integers n for Which ϕ(n)|n – 1. Acta Math Sinica 23, 1915–1918 (2007). https://doi.org/10.1007/s10114-005-0731-1
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DOI: https://doi.org/10.1007/s10114-005-0731-1