If you plot a number as a function of X and the sum of divisors as a function of Y, is anything found in the curve as it goes up in abundance and down in deficiency?
That’s n on the X-axis, and the sum of the divisors (excluding n itself) on the Y-axis. Obviously, there are multiple lines visible “radiating” from the origin. The largest ratio of Y/X for X <= 10000 occurs at X = 5040, with Y = 14304, for a ratio of Y/X = 2.8381. Don’t know if that ratio is bounded or not.
Here’s a zoomed-in portion as well. The line X = Y will be the perfect numbers. ETA: The perfect numbers I found below 10,000 are 12 56 992 16256. So those points that look like they are on the X = Y line mostly aren’t.
I looked more closely at the line near X = Y, where the sum of divisors excluding n (I’ll call it d(n)) is close to n. It turns out that what forms the line is a lot of points where d(n) - n = 12, and to a lesser extent, points where d(n) - n = 56. Intriguing.
Let M be a number divisible by each of 2, 3, …, N (e.g., take M to be N!). Then (the sum of the proper divisors of M)/M is at least 1/2 + 1/3 + … + 1/N.
As the harmonic series diverges, we can thus get arbitrarily large ratios of the desired sort.
If x is perfect and p is a prime not dividing x, then d(px) - px = 2x. Thus, the phenomena you are noticing here are the result of 12/2 = 6 and 56/2 = 28 being the first two perfect numbers.
More generally, let F(n) be the sum of n’s divisors, divided by n. (Thus, F(n) = d(n)/n + 1, so that F(n) - 1 is the slope of the line from the origin to the nth point on the graph given above].
If x and y are coprime, we have that F(xy) = F(x)F(y).
This explains the many approximate “lines” that appear to radiate out from the origin; for every number x, there will be such a “line” whose points correspond to the values y such that F(y) is approximately 1 and y is coprime to x (including every sizable prime).
Or, perhaps better worded, the explanation of the lines is this: the values of F(x) which are achieved early on will nearly-repeat early on as well, as F(xy) for each y for which y is coprime to x and F(y) is nearly 1 [as happens, in particular, when y is a sizable prime].
Ah, wait, an even better way of putting it is this:
From every number x, we get a perfect straight line whose points correspond to the primes apart from those few which divide x. That line is described by the fact that D(px) = (1 + p)D(x) = D(x) + F(x)px for such primes p. [Where D(n) = d(n) + n is the sum of all the divisors of n [Can we just talk in terms of this rather than d(n) since it’s so much better behaved?]]
This line won’t pass through the origin (this was what was throwing me into using scare quotes around the word “line” before, stupidly); rather, its “Y-intercept” will be the sum of (all) the divisors of x. As for its slope, it will be parallel to the line through the origin and the x-th point on the graph.
Isn’t “perfect” rather a depressing name for a number complete unto itself? Rather like the sound of one hand clapping?
Amicable numbers like (220,284) where each is the sum of the other’s divisors seem more romantic: two numbers completing each other.
Even more beautiful perhaps, though outlawed in all states but Utah, are foursomes like
1264460 -> 1547860 -> 1727636 -> 1305184 -> 1264460
I do not like foursomes myself. I’d be happy with a Ménage à trois. Fortunately, since my wife has strong feelings about such things, AFAIK no Ménage-triplet has ever been discovered.
Nah, Utah only allows foursomes consisting of one abundant number and three deficient, not two of each like 1264460 -> 1547860 -> 1727636 -> 1305184 -> 1264460.
It’s no longer considered acceptable to refer to numbers as deficient. The numbers should be referred to as numbers with divisor deficits and/or aliquotically challenged numbers.