14102Variation of perfect numbers

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• Dec 1, 2003
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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 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.
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