On a sum involving the Euler totient function
نویسندگان
چکیده
منابع مشابه
On Square Values of the Product of the Euler Totient and Sum of Divisors Functions
If n is a positive integer such that φ(n)σ(n) = m for some positive integer m, then m 6 n. We put m = n − a and we study the positive integers a arising in this way.
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We consider a multiple arithmetical sum involving the Möbius function which despite its elementary appearance is in fact of a highly intriguing nature. We establish an asymptotic formula for the quadruple case that raises the first genuinely non-trivial situation. This is a rework of an old unpublished note of ours. 2001 Mathematics Subject Classification: Primary 11A25; Secondary 11M06
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متن کاملOn the Ratio of the Sum of Divisors and Euler’s Totient Function I
We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover there exist at most a finite number of solutions to σ(n) = 2 ·φ(n) with Ω(n) ≤ k, and there are at most 22 k+k − k squarefree solutions to φ(n) ∣∣σ(n) if ω(n) = k. Lastly the number of solutions to φ(n) ∣∣σ(n) as x→∞ is of order O (x exp (−1 2log x)).
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ژورنال
عنوان ژورنال: Indagationes Mathematicae
سال: 2019
ISSN: 0019-3577
DOI: 10.1016/j.indag.2019.01.009