In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" ([hyperlinked here](http://utsescholarship.lib.uts.edu.au/dspace/handle/2100/275?show=full)) 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](http://godandmath.files.wordpress.com/2013/07/acms-2013-proceedings.pdf):



> 
> 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.]



On Sorli's Conjecture Re: OPNs (Circa 2003)
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.)


