I learned recently that weekly excess deaths in the US have dipped into negative territory [1], and wondered whether we should start speaking of deficient deaths by analogy with excessive and deficient numbers.
The ancient Greeks defined a number n to be excessive, deficient, or perfect according to whether the sum of the number’s proper divisors was greater than, less than, or equal to n. “Proper” divisor means that n itself isn’t included. Excessive numbers are also called “abundant” numbers.
For example, 12 is excessive because
1 + 2 + 3 + 4 + 6 = 16 > 12,
8 is deficient because
1 + 2 + 4 < = 7 < 8,
and 6 is perfect because
1 + 2 + 3 = 6.
Perfect numbers
Nearly all numbers are excessive or deficient; perfect numbers constitute a very thin middle ground [2]. There are 51 known perfect numbers, all even so far. Euclid proved that if M is a Mersenne prime, then n = M(M+1)/2 is even and perfect. Euler proved the converse: if n is an even perfect number, n = M(M + 1)/2 for some Mersenne prime M.
No one has proved that odd perfect numbers don’t exist, but mathematicians keep proving more and more properties that an odd perfect number must have, if such a number exists. Maybe some day this list will include an impossibility, proving that there are no odd perfect numbers. Currently we know that if an odd perfect number exists, it must have more 1500 digits, its largest prime factor must have more than 8 digits, etc.
Density of excessive numbers
A little less than 1 in 4 numbers are excessive. More precisely, the proportion of excessive numbers less than N approaches a limiting value as N goes to infinity, and that limit is known to be between 0.2474 and 0.2480. See [3]. This means that a little more than 1 in 4 numbers is deficient.
Perfect number density
I said that perfect numbers form a thin middle ground between excessive and deficient numbers. We only know of 51 perfect numbers, but there may be infinitely many of them. Even so, they are thinly distributed, with zero density in the limit. So the density of deficient numbers is 1 minus the density of excessive numbers.
There is a conjecture that there are infinitely many perfect numbers, but that they are extremely rare. As stated above, n is a perfect prime if and only if n = M(M+1)/2 for a Mersenne prime M. A Mersenne prime is a prime number of the form 2p – 1 where p is prime. The number of such exponents p less than a given bound x is conjectured to be asymptotically proportional to
eγ log x / log 2
and the following graph gives empirical support for the conjecture. More on this conjecture here.
If the p‘s are thinly spread out, then the Mersenne primes, 2 raised to the power of these p‘s, are much more thinly spread out, and so even perfect numbers are very thinly spread out.
Related posts
[1] Based on CDC data
[2] Some sources define a number n to be excessive if the sum of its proper divisors is greater than or equal to n, meaning that by this definition perfect numbers are included in excessive numbers.
[3] Marc Deléglise Bounds for the density of abundant numbers
I loved this post! One of my favorite pastimes is to look for super composite and super abundant numbers and think about their prime factorizations.
There are a couple of small typos here…
(Density): “… a little more than *3* is 4 numbers is deficient”
“As stated above, n is a perfect *number* if…” (not perfect prime)
Thanks for your great work!