Perfect numbers
6, 28, 496, 8128, 33550336…
Deficient numbers
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27…
Abundant numbers
12, 18, 20, 24, 30, 40, 43, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102…
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?
(I failed to find an image of the plot online)
By useful I mean something like a predictive pattern…something like the zeroes of the zeta function as it relates to primes.
What do you mean? There’s one right here. 
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.
Sorry, those numbers should be halved: 6 28 496 8128.
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.
This article about divisor functions might be helpful Divisor function - Wikipedia (
Quote:
Originally Posted by Enola Straight View Post
(I failed to find an image of the plot online)
What do you mean? There’s one right here.
Humph…didn’t see the axes properly labled :rolleyes:
Due to 12 being the first abundant number, therefore, multiples of 12 are also abundant.
The ratio is unbounded.
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.
That was a joke. I had just posted it.
Humph.
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.
Love it!
Nah, Utah only allows foursomes consisting of one abundant number and three deficient, not two of each like 1264460 -> 1547860 -> 1727636 -> 1305184 -> 1264460.
I suppose the next would be an amicable octet…no groups of 5,6, or 7.
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.
Edit: how did this get posted in here?
