## The Distribution of Perfection

If we define sigma(n) as the sum of the divisors of n, it's interesting
to consider how the rational numbers sigma(n)/n for n=1,2,.. are
distributed. It's not hard to show that the average of these values
approaches (pi^2)/6. The detailed distribution is also quite
interesting. For example, over the range from n=1 to 100000, exactly
3583 values of sigma(n)/n fall in the interval 1.33 to 1.34, but
only 93 fall in the preceeding interval 1.32 to 1.33. The values
seems to cluster near "low" fractions such as 3/2, 4/3, 7/4, 2, 7/3,
and so on.
Some values of sigma(n)/n occur for more than one n. These seem to
occur in families based on the ordinary Perfect Numbers 6, 28, 496,
etc. If M is a perfect number and p is a prime that does not divide
M, then sigma(pM)/(pM) = 2(p+1)/p. This explains why, for example,
each of the numbers 66, 308, 5456, etc, has a "perfectness" of 24/11.
In general if sigma(n)/n = a/b then n is divisible by b, so we can
tabulate the values of n/b as shown below:
sigma(n)/n n/b
---------- --------------------------------------------------
2/1 6 28 496 8128
8/3 28 90 496 546 8128
12/5 6 28 496 1240 8128
16/5 3(28) 3(496) 3(546) 3(1240)
24/11 6 28 496 8128
32/11 3(28) 3(90) 3(496) 3(546)
48/17 3(28) 3(90) 3(496) 3(546)
52/15 3(28) 3(496) 3(600) 3(1240)
64/23 3(28) 3(90) 3(496) 3(546)
80/29 3(28) 3(90) 3(496) 3(546)
128/47 3(28) 3(90) 3(496) 3(546)
This table suggests that most occurrances of multiple numbers with
the same perfectness are closely related to the sequence of ordinary
Perfect Numbers, plus the intermediate numbers 90, 546, 600, 1240, etc.
QUESTION: What specific value of sigma(n)/n occurs most frequently?

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