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Numbers of the form kf(k)

    https://doi.org/10.1142/S1793042123500586Cited by:0 (Source: Crossref)

    For a function f:, define Nf×(x)=#{nx:n=kf(k) for some k}. Let τ(n)=d|n1 be the divisor function, ω(n)=p|n1 be the prime divisor function, and φ(n)=#{1kn:(k,n)=1} be Euler’s totient function. We prove that

    (1)Nτ×(x)x(logx)1/2;(2)Nω×(x)=(1+o(1))xloglogx;(3)Nφ×(x)=(c0+o(1))x1/2,
    where c0=1.365.

    AMSC: 11N37, 11N64, 11B83

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