Big "R" & The Hunt For Phat Numbers

[Turtle]

Define S as the sum of an integer's divisors (as defined by perfect numbers)
Definition: R (say "big r") is the ratio S(n)/n. That is, big R is the ratio of the sum an integer's divisors to itself.

Question(s): Is there an upper limit to R? Can we even know if there is limit?

Extra: I have found that for the first few million integers, R's above 3.0 are infrequent. I call these "phat numbers" as they are excessively abundant. In my current hunt for a "phatest" number under big R, I have this phat number:

n = 10,810,800 S(n) = 40,852,560 480 proper divisors R = 3.7788656


Challenge: Find a phater number than I have. Then post it here.

[Turtle]

My hunt is ongoing from a run I began about 2 weeks ago. (See thread "Strange Numbers" for a more detailed description of the general experiment environment) On a 200 Mghz machine, I have now factored the first 13,075,089 integers; the current rate is approx. 341 integers per minute. Here are the last 5 Phat numbers:

Phat Numbers

#Of Divisors Integer Sum R
360 3,603,600 16,790,592 3.6594
384 4,324,320 20,321,280 3.6993
432 7,207,200 26,915,616 3.7345455
448 8,648,640 32,316,480 3.7365967
480 10,810,800 40,852,560 3.7788656

Addendum: An interesting side note: Did you know numbers of the form abc,abc always divide by 7, 11, 13, 77, 91, 143, & 1001?

[Turtle]

No new Phatest number found at this post. 14,081,000 integers now factored & running at a rate of approx 330 per minute. You cant get past that Big O, when you want the Big R.

[Turtle]

Here's the update. At this post, 15,017,000 integers factored & running at the rate of approx. 330 per minute. In looking at the last update I see I also posted 330/min & I beleive this is underestimated now. I am simply using my watch to time the real-time display.
Now I do have a new Phat Number for you, but it isn't the Phatest. It does however exibit that abc,abc form I mentioned & that the other Phat Numbers have as well. Curiouser & curiouser.

#Of Divisors Integer Sum R
540 14,414,400 54,372,864 3.7721212

[sanctus]

keep on posting your update, even if nobody answers, I'm (therefore at least one person) interested in your search of the phatest number.
Maybe your are about to find a new constant (turtle's constans) to add to eulers and so on.

by the way your number isn't of the form abc,abc but there are some zeros as well.

[Turtle]

Roger Sanctus. You make a good point on the form of the last number;it only contains the pattern abc,abc. I didn't specifically check for the divisors of abc,abc I listed, but now I will. Of course, adding the ending zeros indicates the number divides by 2, 4, 5, 10, & 100.
Although I noticed Phat numbers a number of years ago, it wasn't until I started this thread that I collected them in any sort of order. The kind of order that seems to reveal in some degree a consistant pattern abc,abc.
This is exactly the kind of participation I hoped for by conducting the experiment "live" as it were. Observations I have overlooked & questions I neglected to ask from the collective conciousness of Hypography. Truly a unique opportunity for discovery. Thanks Sanctus.

[Turtle]

___As I have developed this in reclusion, it is always in the back of mind that I'm simply deluded in seeing these patterns. So now, by way of Hypography, I have described some of the patterns I think i see, & in this thread I called the pattern I saw "Big R". Well, I can hardly google that, since it is my own term. Now of course I realize I, nor even we, can in a lifetime read everything in math that pertains to this; not to mention some dusty ancient scroll or tome undiscovered.
___After the Hypography Live Chat today, I googled "perfect numbers" for the umpteenth time, & stumbled onto a page which desribes this very situation of "Big R". I have the link below, but first I want to translate my interpretation.
___The page defines o(n)/n as multiplicity & a couple other interchangeable terms. This is "Big R" Now as I understand the article, they are only looking for integer values of R, that is no fraction. In my hunt here, I have only found R's < 4, but it appears to me by the link that at least one integer of R =11.0 has been found. I tried to look at the lists linked to the article, but this machine doesn't recognize the format.
___No doubt a number with R >=11 is bigger than my hardware/software can handle. This is partly why I shared my delusion. Nonetheless, if somebody else saw it, & they have, & spent a bunch of time on it, & they did, then at the very least it is a shared delusion. I feel much better.
___They do not address however my delusion of a limit to R or whether such is knowable. Further, someone once suggested my Strange Numbers were "pluperfect" & as this article defines that, I see Strange Numbers are not pluperfect. Still a unique delusion. followed in the other thread.

http://wwwhomes.uni-bielefeld.de/achim/mpn.html

[Lazlo Toth]

This is all very facinating! Is there a "thinnest" Phat number? I know you are looking for large numbers of divisors, but I would be quite curious to know what the distribution curve would look like? Is it monotonically decreasing? I will have to pull out my C++ compiler and work on this!

Lazlo

[Turtle]

___ A 'thinnest phat number'...hmmmm. I note that on the page I posted above, they include the number itself in the sum, as in the example they give for 120 being '3-perfect'. This is not incorrect per se, however the experiments I have described do not count the number itself in the sum. The 'standard' definition of Perfect Numbers, does not include the number itself, however including it only changes the Big R in the integer portion. By their description , a Perfect Number is one whose divisor sum is exactly twice itself; in the standard definition, a Perfect Number is one whose divisor sum equals itself.
---So, by the standard definition & my experiments here, a 'thinnest' Phat number(actually we should be saying 'thinnest abundant' number) might be a number that is > 1 (abundant) by the least amount. Does that sound right? Hmmm.....

[Qfwfq]

Originally Posted by Turtle

Addendum: On interesting side note: Did you know numbers of the form abc,abc always divide by 7, 11, 13, 77, 91, 143, & 1001?

This is quite trivial to explain: such a number is exactly 1001 multiplied by abc. 7, 11, 13, 77, 91, 143 are the divisors of 1001 and hence also of the given number.

The extra zeroes add the prime factors 2 and 5 and all the 'abc' values you list are multiples of 3, thus completing the primes up to 13. On the basis of a bit of reckoning I did after finding this thread a couple of hours ago I do not find it surprising that your list members all exhibit this. I'm on spare time at work now and my reckonings may have arrived at the limits of predictability but, if I have time to think over the weekend, I'll tell you if I find anything else interesting next week.

A few easy things to work out:

1) If a, b, c... are the multiplicities of the prime factors of a given N, the number of its proper divisors is 1 less than (a + 1)(b + 1)( c + 1)...

2) R, as you define it, may be written as the number of divisors times the average of these, all divided by the given N; it is hence less than the term with the max divisor in lieu of the average; the max proper divisor is n divided by its smallest prime factor.

3) For given numbers of similar size, large PFs hike up the average divisor but hike down the sum of the multiplicities (i. e. a + b + c + ...). For the same sum, however, diversity of primes raises the number of proper divisors according to the computation given above(1). Were it not for the latter point, the Phatest Ns would be powers of 2 but, instead, these have R approaching 1 for increasing exponent.

I hope you might find these considerations helpful to your quest. Since you are kneading the dough, try working out R for the following three candidates, a comparison of the three results could provide clues, if I don't find anything more useful during the next few days:

1853628416640000, 23538138624000, 3977945427456000

What remains to analyse is computation of the above mentioned average divisor and how to handle it in the aim of large R.