## Variation of perfect numbers

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• I ve been looking into a couple of minor variations on the perfect number ... Of course, a perfect number is equal to the sum of its proper factors. What if
Message 1 of 2 , Dec 1, 2003
I've been looking into a couple of minor variations on the "perfect
number"... Of course, a perfect number is equal to the sum of its
proper factors.

What if we exclude its prime factors? What numbers are equal to the
sum of their non-prime proper factors? (Note that the factor one
is included, since 1 is not prime.) I can find five:

42
1316
131080256
72872313094554244192
37778715690312487141376

Note that all of these except for 72872313094554244192 are of a
special form:

q = 2^p-1
r = 2^(2*p)-2^(p+1)-1

N = 2^(p-1)*q*r

where p,q,r are all prime. In this case, we get solutions for
p=(2,3,7,19).

That leaves the "oddball" number:

72872313094554244192 == 2^5 * 109 * 151 * 65837 * 2101546957

Which seems to be just a stroke of random luck.

Perhaps even more interesting, what numbers are equal to the sum
of their composite proper factors? Same problem as above,
except that the factor 1 is not counted. Despite searching
long and hard, I haven't found a single example. I can't think
of any reason why such numbers shouldn't exist, nor why they
should be so scarce.

Has anybody ever heard of other research on these two types of
numbers? I will continue my search for now, but if I'm treading
on already explored ground, I would love to hear about it. A
Google search turned up nothing of interest. In particular,
the number 131080256 doesn't show any hits, despite its rather
"rare" nature.
• I don t know of any work on the numbers you mention, but the function: sum of nonprime factors of n is listed in the OEIS: ID Number: A023890 URL:
Message 2 of 2 , Dec 1, 2003
I don't know of any work on the numbers you mention, but the function:
sum of nonprime factors of n is listed in the OEIS:

ID Number: A023890
URL: http://www.research.att.com/projects/OEIS?Anum=A023890
Sequence: 1,1,1,5,1,7,1,13,10,11,1,23,1,15,16,29,1,34,1,35,22,23,1,55,
26,27,37,47,1,62,1,61,34,35,36,86,1,39,40,83,1,84,1,71,70,
47,1,119,50,86,52,83,1,115,56,111,58,59,1,158,1,63,94,125,
66,128,1,107,70,130,1
Name: Sum of nonprime divisors of n.

On Mon, 1 Dec 2003, Jack Brennen wrote:

>
> I've been looking into a couple of minor variations on the "perfect
> number"... Of course, a perfect number is equal to the sum of its
> proper factors.
>
> What if we exclude its prime factors? What numbers are equal to the
> sum of their non-prime proper factors? (Note that the factor one
> is included, since 1 is not prime.) I can find five:
>
> 42
> 1316
> 131080256
> 72872313094554244192
> 37778715690312487141376
>
> Note that all of these except for 72872313094554244192 are of a
> special form:
>
> q = 2^p-1
> r = 2^(2*p)-2^(p+1)-1
>
> N = 2^(p-1)*q*r
>
> where p,q,r are all prime. In this case, we get solutions for
> p=(2,3,7,19).
>
> That leaves the "oddball" number:
>
> 72872313094554244192 == 2^5 * 109 * 151 * 65837 * 2101546957
>
> Which seems to be just a stroke of random luck.
>
> Perhaps even more interesting, what numbers are equal to the sum
> of their composite proper factors? Same problem as above,
> except that the factor 1 is not counted. Despite searching
> long and hard, I haven't found a single example. I can't think
> of any reason why such numbers shouldn't exist, nor why they
> should be so scarce.
>
> Has anybody ever heard of other research on these two types of
> numbers? I will continue my search for now, but if I'm treading
> on already explored ground, I would love to hear about it. A
> Google search turned up nothing of interest. In particular,
> the number 131080256 doesn't show any hits, despite its rather
> "rare" nature.
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

------------------------------------------------------------
W. Edwin Clark, Math Dept, University of South Florida,
http://www.math.usf.edu/~eclark/
------------------------------------------------------------
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