The earliest result appears to be $$\frac{1}{2}<\sum_{p\mid N}\frac{1}{p}<2 \log \left (\frac{\pi}{2} \right ),$$
by Perisastri. Other authors have sharpen this result. For example, for a perfect number $N\equiv 0 \mod 3$, the sum has $$\sum_{p\mid N}\frac{1}{p}=\frac{1}{3}+ \cdots+\frac{1}{p},$$ see Ribenboim, New Book of Prime Records, p. 101, for a discussion and other references.
Perisastri, M. On the non-existence of odd perfect numbers of a certain form. Math. Student 28 1960 85–86 (1962). MR0142494.
