On Wed, 1 Sep 1999, Spike Jones wrote:
> With every Mersenne number there is an associated perfect
> number, the sum of whose factors exactly equal the number.
A number is perfect iff the sum of the positive divisors, including
one and excluding the number itself, is equal to the number.
> I discovered a fascinating thing today, for which I must introduce
> some new terminology.
>
> If a number is greater than the sum of its factors, let it be a cold number.
> If a number is less than the sum of its factors, let it be a hot number.
If s(n) is the sum of the positive divisors of n including one
then a perfect number has s(n) = 2n. From Chapter XIV of Nicomachus
If s(n) > 2n then n is abundant and for s(n) < 2n then n is deficient.
> Odd numbers are all cold, for instance, and the first hot number is 12.
945 is odd and abundant.
> Nowthen, I found that the ratio of cold numbers to hot numbers is
> always about 3. Even when you get up to large numbers [I checked
> them all up to about 100,000] the ratio seems to stay right around
> 3 colds to every hot.
This is known. See, for example, Deleglise, M. ``Bounds for the
Density of Abundant Integers.'' Exp. Math. 7, 137-143, 1998 at
http://www.expmath.org/restricted/7/7.2/deleglise.ps.gz
He gives the result 0.2474 < A(2) < 0.2480 where A(2) is the density
of abundant numbers (he includes perfect numbers as abundant).
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