Iterating the Sum of Möbius Divisor Function and Euler Totient Function
نویسندگان
چکیده
منابع مشابه
Iterating the Sum-of-Divisors Function
1991 Mathematics Subject Classi cation: 11A25, 11Y70 Let 0(n) = n and m(n) = ( m 1(n)), where m 1 and is the sum-of-divisors function. We say that n is (m; k)perfect if m(n) = kn. We have tabulated all (2; k)-perfect numbers up to 109 and all (3; k)and (4; k)-perfect numbers up to 2 108. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any ...
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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: Mathematics
سال: 2019
ISSN: 2227-7390
DOI: 10.3390/math7111083