TITLE: Conjecture on Odd Perfect Numbers
QUESTION [6 upvotes]: I'm a grad student in mathematics and I've been working with a very gifted high school student (likely the smartest high school student I've ever met) on problems he's brought up and some competition math problems. This student has developed an interest in perfect numbers and the question regarding existence of odd perfect numbers. He has come up with a conjecture about odd perfect numbers, but I have not studied number theory and hence am not necessarily aware of well-known results of the field. So, here we are. 
His idea: 
Suppose $N \in \mathbb{N}$, with prime decomposition $N = p_1^{q_1}\cdots p_n^{q_n}$ Define $\tilde{N} = p_1\cdots p_n$.
Conjecture: If $N$ is an odd perfect number, then the sum of reciprocals of all factors of $\tilde{N}$ (excluding 1, including $\tilde{N}$) is less than 1.

Q. Does this conjecture appear to be equivalent to something that has already been established? If this conjecture is true, does it appear to have any obvious implications?

REPLY [3 votes]: 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.