A'natural number n is called k-pcrfect if σ(n) = kn. In this paper, we show that for any integers r ≥ 2, and k ≥ 2, the number of odd k-perfect numbers n with ω(n) ≤ r is bounded by (([4'log_32]+r)/r) ∑_(i=1)~r (([kr/2])/i), which is less than 4~(r~2) when r is large enough.
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