# Perfect, abundant, and deficient numbers **URL:** \ **Category:** Factual Questions\ **Created:** [October 23, 2013, 1:20am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990 "2013-10-23T01:20:29Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 23, 2013, 1:20am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/1 "2013-10-23T01:20:29Z")
Perfect numbers > **[Perfect number](https://en.wikipedia.org/wiki/Perfect_number)** > > In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. > The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including it... 6, 28, 496, 8128, 33550336… Deficient numbers > **[Deficient number](https://en.wikipedia.org/wiki/Deficient_number)** > > In number theory, a deficient number or defective number is a number n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient. > Denoting by σ(n) the sum of divisors, the value 2n − σ(n) is called the number's deficiency. In terms of the aliquot sum s(n), the deficiency is n − s(n). > The first few d... 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27… Abundant numbers > **[Abundant number](https://en.wikipedia.org/wiki/Abundant_number)** > > In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. > A number n for which the sum of divisors σ(n) \> 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) \> n. > Abundance is the ... 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) ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 23, 2013, 10:05am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/2 "2013-10-23T10:05:05Z")
By useful I mean something like a predictive pattern…something like the zeroes of the zeta function as it relates to primes. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [October 23, 2013, 9:53pm UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/3 "2013-10-23T21:53:48Z")
> [@Enola\_Straight](#): > > (I failed to find an image of the plot online) What do you mean? There’s one [right here](http://i.imgur.com/1KZn3I3.jpg). 🙂 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](http://i.imgur.com/YD2nZhk.jpg) 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. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [October 23, 2013, 10:01pm UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/4 "2013-10-23T22:01:08Z")
Sorry, those numbers should be halved: 6 28 496 8128. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [October 24, 2013, 12:27am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/5 "2013-10-24T00:27:13Z")
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. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [October 24, 2013, 12:34am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/6 "2013-10-24T00:34:58Z")
This article about divisor functions might be helpful [Divisor function - Wikipedia](http://en.wikipedia.org/wiki/Divisor_function) ( ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 24, 2013, 1:00am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/7 "2013-10-24T01:00:16Z")
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: > [@ZenBeam](#): > > 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. Due to 12 being the first abundant number, therefore, multiples of 12 are also abundant. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 24, 2013, 1:32am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/8 "2013-10-24T01:32:42Z")
> [@ZenBeam](#): > > What do you mean? There’s one [right here](http://i.imgur.com/1KZn3I3.jpg). 🙂 > > 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. 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. ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [October 24, 2013, 1:49am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/9 "2013-10-24T01:49:23Z")
> [@Enola\_Straight](#): > > 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: That was a joke. I had just posted it. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 24, 2013, 2:07am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/10 "2013-10-24T02:07:38Z")
Humph. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 24, 2013, 2:36am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/11 "2013-10-24T02:36:40Z")
> [@ZenBeam](#): > > 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. 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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 24, 2013, 3:08am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/12 "2013-10-24T03:08:00Z")
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). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 24, 2013, 3:15am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/13 "2013-10-24T03:15:14Z")
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]. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 24, 2013, 4:27am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/14 "2013-10-24T04:27:13Z")
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. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [October 24, 2013, 11:28am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/15 "2013-10-24T11:28:05Z")
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. ---
**Author:** ![JKellyMap](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jkellymap/32/15880_2.png) [@JKellyMap](https://boards.straightdope.com/u/JKellyMap)\ **Post date:** [October 24, 2013, 10:56pm UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/16 "2013-10-24T22:56:03Z")
> [@septimus](#): > > 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… Love it! ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 24, 2013, 11:58pm UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/17 "2013-10-24T23:58:30Z")
Nah, Utah only allows foursomes consisting of one abundant number and three deficient, not two of each like 1264460 -\> 1547860 -\> 1727636 -\> 1305184 -\> 1264460. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 25, 2013, 12:20am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/18 "2013-10-25T00:20:40Z")
> [@septimus](#): > > 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. I suppose the next would be an amicable octet…no groups of 5,6, or 7. ---
**Author:** ![robert\_columbia](https://avatars.discourse-cdn.com/v4/letter/r/e79b87/32.png) [@robert\_columbia](https://boards.straightdope.com/u/robert_columbia)\ **Post date:** [October 25, 2013, 12:38am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/19 "2013-10-25T00:38:12Z")
> [@Enola\_Straight](#): > > …Deficient numbers… 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. ---
**Author:** ![Cerealbox](https://avatars.discourse-cdn.com/v4/letter/c/7ab992/32.png) [@Cerealbox](https://boards.straightdope.com/u/Cerealbox)\ **Post date:** [October 25, 2013, 3:02am UTC](https://boards.straightdope.com/t/perfect-abundant-and-deficient-numbers/671990/20 "2013-10-25T03:02:52Z")
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