In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler prime $p$ for an odd perfect number $N = (p^k)(m^2)$ is one (i.e. we can drop $k$).
Assuming Sorli's conjecture is true, does anyone know if there exist (any) "effective" results (pardon my use of the term, I just could not think of a better word) in the literature, particularly with respect to relations between the Euler prime $p$, the exponent $k$ and the number $\sqrt{\frac{N}{p^k}}$? I have, so far, only been able to get hold of Paolo Starni's article titled "Odd Perfect Numbers: A Divisor Related to the Euler′s Factor".
In particular, note that Sorli's conjecture implies the following relations:
$$I(p^k) = I(p) = \frac{p+1}{p}$$
$$I(m^2) = \frac{2}{I(p)} = \frac{2p}{p + 1}$$
which, in turn, gives the (trivial) algebraic identity:
$$\frac{1}{p} = \frac{1}{p+1} + \frac{1}{p}\left(\frac{1}{p+1}\right)$$
where $p$ is the Euler prime (i.e. $p^k$ is the Euler's Factor) and $$I(x) = \frac{\sigma(x)}{x}$$ is the abundancy index of $x$.
[Edit (September 18 2013) – Per Professor Beasley's paper titled "Euler and the Ongoing Search for Odd Perfect Numbers" from this hyperlink:
Before proceeding with Euler’s proof, we pause to note that his result was not quite what
Descartes and Frenicle had conjectured, as they believed that $k = 1$, but it came very
close. In fact, current research continues in an effort to prove $k = 1$. For example,
Dris has made progress in this direction, although his paper refers to Descartes’ and
Frenicle’s claim (that $k = 1$) as Sorli’s conjecture; Dickson has documented
Descartes’s conjecture as occurring in a letter to Marin Mersenne in 1638, with
Frenicle’s subsequent observation occurring in 1657.
End Edit.]
Best Answer
As far as I know, there are no such effective bounds. In fact, even if $p=5$ and $k=1$, there are no known effective bounds on $N$. (There are bounds on $N$ in terms of the number of distinct factors.)