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.