1. Numbers of the form kf(k).
- Author
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Gabdullin, Mikhail R., Iudelevich, Vitalii V., and Luca, Florian
- Subjects
- *
PRIME numbers - Abstract
For a function f : ℕ → ℕ , define N f × (x) = # { n ≤ x : n = k f (k) for some k }. Let τ (n) = ∑ d | n 1 be the divisor function, ω (n) = ∑ p | n 1 be the prime divisor function, and φ (n) = # { 1 ≤ k ≤ n : (k , n) = 1 } be Euler's totient function. We prove that (1) N τ × (x) ≍ x (log x) 1 / 2 ; (2) N ω × (x) = (1 + o (1)) x log log x ; (3) N φ × (x) = (c 0 + o (1)) x 1 / 2 , where c 0 = 1. 3 6 5 .... [ABSTRACT FROM AUTHOR]
- Published
- 2023
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