About Descartes numbers I just have a quick question regarding the definition of Descartes numbers, otherwise known as spoof odd perfect numbers.
In this Wikipedia page, Descartes numbers are defined as follows:

A Descartes number is defined as an odd number $n = mp$ where $m$ and $p$ are coprime and $2n = \sigma(m)\cdot(p+1)$.

My question is:  Is the divisibility constraint $\gcd(m,p)=1$ necessary?  I tried perusing Banks et. al.'s paper on the subject, but they do not seem to make that assumption.  I also checked Dittmer's "Spoof odd perfect numbers" and he does not make that supposition either.  Lastly, I tried searching for a page on Descartes numbers in MathWorld@Wolfram in the hopes of comparing the definition in Wikipedia, but there was none.
Can anybody verify and confirm if the definition of Descartes numbers, as stated in Wikipedia, is correct?  
 A: Per an e-mail response from Professor Bill Banks:
From: Banks, William D.
Date: Sun, Mar 8, 2015 at 9:23 AM
Subject: RE: Your paper titled "Descartes Numbers"
To: Jose Arnaldo Dris
Dear Arnie,
To the best of my knowledge, the term "Descartes number" was first coined by me and first appeared in my paper with Guloglu, Nevans and Saidak.  However, I could easily imagine
that the terminology had been used before, though I had never seen it.
Because of that, I'd say the Wikipedia definition should be changed; the coprimality condition need not be included.  Another thing I dislike about the Wikipedia entry is the use of "$p$" in the definition, since $p$ may well be composite, as in Descartes'
example.
Cheers,
Bill
From: Jose Arnaldo Dris
Sent: Friday, March 06, 2015 1:05 PM
To: Banks, William D.
Subject: Re: Your paper titled "Descartes Numbers"
Hi again Professor Banks,
Additionally, I would like to refer you to a question that I've recently posted over at Math@StackExchange - About Descartes numbers.
Would you be able to confirm if the definition for Descartes numbers in Wikipedia is correct?
Thanks,
Arnie Dris
