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. :cup:

[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

:cup:

[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]

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.

[Turtle]

___Very interesting Q! The numbers you listed are too large for me to factor with my current hardware/software setup. I have an upper limit on integers of just over 2 billion(do you use the same billion we do over in Italy; 2,000,000,000?). If you looked at the page on "multiperfect numbers" I posted above, they list 14,182,439,040 as the first "5-fold perfect number; they count the number itself in the sum & I don't in my experiment. What this means is that 14182439040 under my system has a Big R of 4.0000.

___I have stopped my run to use the machine for some other projects, however at last count I factored 17,086,000 integers & the program had slowed to 325 integers per minute.

___ Here is a Phat number that seems to have a pattern, but it isn't abc,abc:

 

#Of Divisors_____ Integer_____ Sum ___________Big R

512___________17,297,280____64,955,520___3.7552447552

:D

[Turtle]

___In story posts #8 & #9 of this thread, the idea of a thinnest number came up. On further consideration of this, I have some observations. In the related Hypography thread "Strange Numbers...", it is noted that a Strange Number is one whose sum of divisors excedes it by exactly 12 & further I proposed the set of Strange Numbers is infinite.

___Now since that difference of 12 doesn't change(by definition), then as the size of the Strange Numbers increase, Big R must decrease. So I'm suggesting that the Thinnest Abundant Numbers are the Biggest Strange Numbers.

___Now in addition, there may be numbers abundant by less than 12, ie. abundant by 1, 2, 3, 4, 5,6 ,7 8, 9, 10, & 11. I have never thought before to look. Still, if they exist, they may not qualify for Thinnest depending on their Big R.

___I love this stuff! :o

[Qfwfq]

___ Here is a Phat number that seems to have a pattern, but it isn't abc,abc:

 

#Of Divisors_____ Integer_____ Sum ___________Big R

512___________17,297,280____64,955,520___3.7552447552

:(

17,297,280 = 1001 * 1728. It's only a bit less obvious cuzz 1728 isn't less than 1000. :(

 

Don't worry Turtle! I found a spot of time Saturday eve to write a bit of C++ using a thing I had done some time ago for handling rational-value arithmetic. I have considered an LGPL release of it but it still maybe needs a bit more work on it and I won't publish it quite yet. However, type double seems to handle the values I was using with reasonable precision and with faster execution times, my Rational class is precise but it can be very heavy; better for re-checking the best finds, perhaps. Also, a couple of the things at http://directory.fsf.org/science/math/ could be handy for the job too.

 

My approach is to choose powers of the smallest primes and multiply them up, as opposed to factoring which is slower! The values I posted Friday are:

 

1853628416640000 = 2^10 * 3^10 * 5^4 * 7^3 * 11 * 13

 

23538138624000 = 2^12 * 3^8 * 5^3 * 7^2 * 11 * 13

 

3977945427456000 = 2^12 * 3^8 * 5^3 * 7^2 * 11 * 13^3

 

and I computed these Friday on the calculator in Windows accessories, in scientific mode (from the View menu). ;)

 

edit: I mean that I computed the three Ns, not the three Rs :( , by calculator!!!

 

Saturday I tried similar things including instances with 11^2 and 13^2 which seem to increase R more than incrementing the power of 2, as well as the sequence 2^n * 3^n * 5^n * 7^n * 11^n * 13^n that gives great values with n going 5 and above (careful: the numer of divisors to add up increases drammatically with n :xx: ). No difficulty getting values above 4!

 

I currently can't state a definite recipe but it seems better to use the smallest consecutive primes with powers greater than 1. If I could dedicate the time to it, I would work on it in terms of the distribution of these powers... :cup:

[Kent]

[

Define S as the sum of an integer's divisors (as defined by perfect numbers)

 

Hmm....well I don t think that is the definition of what a perfact numbers suppost to be. Since 6 is a common perfect number, and it divisor is 1,2, 3, and 6. the sum of it s divisor is 1+2+3+6=12. 12 do not equal 6. A number is perfact if the sum of it oliquot parts is equal to itself.

 

Definition: R (say "big r") is the ratio S(n)/n. That is, big R is the ration 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?

 

suppost s(n) is the sum of the oliquot part of n.

 

I am not sure if there are infinite number of perfact number, but if there are

 

then s(n)/n =1 at some unique value of n. the lim(n->infintity) S(n)/n do not exist , because it is not a continuous graph. At least i cannot imagine a limited where the graph jump.

 

Suppose it is the sum of the divisors of a number divide by the number.

 

suppose( n) is the sum of the oliquot parts of n

 

then: the ratio will look something like:

 

(s(n)+n)/n = 1 + (s(n)/n) >1

 

Again, it is not a continuous function. The limited do not exist.

[Qfwfq]

This morning I said: "I would work on it in terms of the distribution of these powers..."

 

It struck me later that one could do this with a great saving of execution time, for all powers, up to n, of the first p primes, if the program can allocate an array of double values with p indices each up to n (1 + n values). This means several megabytes but it can be done if RAM is sufficiently available and not bogged down by other stuff. Avoid that RAM being swapped!!!

 

The great saving is because each of the products will be an N-value and, except for the largest one, it will also be a divisor for other Ns. In fact:

 

2^a * 3^b * 5^c * 7^d * 11^e * 13^f

 

is a divisor of any N having no lower exponents and at least one higher one. Note: choosing the first p primes is equivalent to choosing at the most p, because an exponent of zero is equivalent to the prime being left out. :xx:

 

#define MAX_EXP 9

#define NUM_PRMS 6 // these choices require an 8 meg array

 

unsigned char primes[NUM_PRMS] = {2, 3, 5, 7, 11, 13};

 

double products[MAX_EXP][MAX_EXP][MAX_EXP][MAX_EXP][MAX_EXP][MAX_EXP];

 

main(){

// first, iterate 6 levels filling up products[].

// Shouldn't take too tooo loooooong...

//

// then, iterate again, for each N, add up its divisors and then divide.

// Will take loooooooooooooooonger...

//

// have fun!!!! :(

}

 

Hint: if type double's precision turns out troublesome, then multiplication is no longer associative... plan the order in products carefully! :cup:

[Qfwfq]

[

I am not sure if there are infinite number of perfact number, but if there are

 

then s(n)/n =1 at some unique value of n. the lim(n->infintity) S(n)/n do not exist , because it is not a continuous graph. At least i cannot imagine a limited where the graph jump.

:cup:

 

Apart from the fact that a non continuous function may have a limit, the point of Turtle's question is whether the expression has a finite supremum. It might, though as far as I can see I don't tend to think so. BTW, according to which topology on the natural numbers would you consider a function of n to be continuous? :(

 

The point about including N itself, along with its proper divisors, is a quite easy matter and Turtle had already more or less pointed it out.

[Turtle]

___After reviewing the last posts by Kent &Qwfwq & the link I posted in #7, I too believe there is no limit to Big R. I suspect Qwfwq's method is sufficient for finding Big R's, but it may not be necessary, ie. it may not find all big R's. Just a guess coming from the idea of using Mersenne Primes to find perfect numbers & that it finds some but not all.

___That settled, what do you think of my assessment of "Thinnest" Abundant numbers? I did add a trap for numbers abundant by less than 12 to my current factoring run, which is at about 18 million now. I should restart the run at 1 & look, but it's taken me a month on my 200MGHZ to get this far. Still, I believe the 'Thinnest ' Abundant numbers are the Strange numbers I identified (see thread 'Strange Numbers...').

___A final observation on factoring. Part of the reason I set up the experiments as I have, is to see flashing by on the screen, what is really going on as decribed by shortcuts like prime factorizations, etc. Yes factoring is hard; that is why I chose to do it. All of my observations stem from watching what is really going on, in the knowledge of what the expressions say is going on. Anyway, keep the suggestions, expressions, & observations coming. :(