- sorry that I've disappointed you in the past. the fact that 22021 isn't an odd-perfect number is because it has four factors total including the '1'.

the summation would be even-- 23622; not worring about it being abundant, it has to have and 'even' number of factors plus 1 so that 'odd' summation equals 'odd' product. 22021 disobeys the definition from that standpoint.

--- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:

>

> Right. I'm mentioned this to him just about every time that

> he posts a "proof" -- and he has never addressed the issue.

>

> Every one of his proofs -- as far as I can tell -- claims that

> such a construct can't exist. My counterexample shows that

> such a construct can in fact exist, and the only reason it's

> not an odd perfect number is because 22021 is divisible by 19.

> But none of his arguments ever explain why such a construct

> can exist with one of the factors composite, but cannot exist

> with all of the factors prime.

>

>

>

> On 7/11/2011 5:44 PM, djbroadhurst wrote:

> >

> >

> > --- In primenumbers@yahoogroups.com,

> > Jack Brennen<jfb@> wrote:

> >

> >> According to your proof, why isn't N = 3^2*7^2*11^2*13^2*22021

> >> an odd perfect number?

> >

> > Jack's point is well made; he has shown that

> > sigma(N)/(2*N) = sigma(22021)/(1+22021).

> > If 22021 were prime, N would be an odd perfect number.

> > Of course, 22021 = 19^2*61 is not prime.

> > However Bill did not mention decomposition into /primes/

> > in his vague remarks; so he might not detect that N is abundant.

> >

> > David

> >

> >

> >

> > ------------------------------------

> >

> > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com

> > The Prime Pages : http://www.primepages.org/

> >

> > Yahoo! Groups Links

> >

> >

> >

> >

> >

> - Thx, I will rework it.

________________________________

From: Tom Hadley <kctom99@...>

To: Mathieu Therrien <mathieu344@...>

Cc: "primenumbers@yahoogroups.com" <primenumbers@yahoogroups.com>

Sent: Thursday, July 14, 2011 4:41:26 PM

Subject: Re: [PrimeNumbers] Post-Cartesian Puzzle

Mathieu Therrien <mathieu344@...> wrote:

>

I think you have misunderstood the sigma() function. In Pari-GP, sigma(x) is the sum of the divisors of x. So sigma(9) = 1+3+9 = 13.

>If I understood the purpose of Sigma(N/M) = N/P_1 = M*P_2 for P = P_1 * P_2 correctly,

>

>then Many solution are possibles as long as (M+1) is divided by 2 only once

>

>for example u have m=5 ; N = P_2 * m * 3 = 15*P_2 and as long that P_2 is odd

>

>So m=5 and N=45 is 1 solution

>

>

The puzzle is: Find a pair of odd integers (N,m) with m|N,

sigma(N/m)*(1+m) = 2*N, and bigomega(m) = 1.

The proposed solution, N=45, m=5 doesn't work, since

sigma(45/5)*(1+5) = sigma(9)*6 = 13*6 = 78, which is not 2*N=2*45=90.

Tom Hadley

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