Big "R" & The Hunt For Phat Numbers

[Kent]

: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.

 

 

Well, i just saw the first post, other than that,i did not read anything on this thread. If i am wrong i am wrong.

 

non continuous function may have a limit

 

How is it possible? Actually i don t even recalled the specific definition in calculus, but i just think it is a weird looking function. one that is not predictable. As n approach infinity, you go go go and, got back to one. What the hell is that? please explain.

 

the point of Turtle's question is whether the expression has a finite supremum

 

What is a supremum?

 

according to which topology on the natural numbers would you consider a function of n to be continuous

 

i am still in calulus, but do amuse me.

[sanctus]

How is it possible? Actually i don t even recalled the specific definition in calculus, but i just think it is a weird looking function. one that is not predictable. As n approach infinity, you go go go and, got back to one. What the hell is that? please explain.

 

Take a function that is zero for negative x, 1 for positive (and x=0) x. This function (called heavyside function) is not continous in zero, but the limit at plus infinity is one and the limit at minus infinity is zero.

Or take a function that changes it's value in a non continuos way in the interval from -1000 to 1000 for example, then the function is zero, the limit at +/- infinity is zero.

 

 

A supremum is the smallest number that is bigger than every member of a given set of number (actually if the set of numbers are for example integers, the supremum hasn't got to be an integer)

[Qfwfq]

All right Kent, if you want to learn that's a great thing, but some of your questions are a bit outside the topic of this thread. Also, as I said in introductions, I can't spend a great deal of time posting, otherwise I'd be even more willing to help you. Before I got this off sanctus posted, but anyway...

 

Supremum: this is a bit more relevant to Turtle's topic in this thread. It's a Latin word often abbreviated sup, used to indicate the upper limit of all the values of a set, even when it isn't actually a value belonging to the same set. If it does belong to the set, it is called the maximum. The supremum isn't necessarily the lim(x ---> something). A majorant of the set is a value greater than any in the set; the supremum is therefore the "smallest majorant" of the set.

 

Back to Turtle's Topic, I agree that there is probably not a finite upper limit (sup) but I haven't currently worked out a proof. I also agree that the limit lim(n ---> infinity) is not defined (these two "limits" are two different things) but my argument wouldn't be that of continuity (which has scarce meaning over the natural numbers). Roughly:

 

1)For any n (however large) there will be primes P greater than n; for these R = 1/p, which approaches zero.

 

2)There will also be numbers greater than the same n having diverse prime factors with a large total of multiplicities, hence a great number of divisors, hence R distinctly greater than zero.

 

3)Hence, for increasing n, the R values in the range above n do not "narrow around" some value L (this is where my argument is a little rough!), i. e. lim(n ---> infinity) is not defined. (this does :hihi: not mean that L is infinite, it means there is no such thing as that limit)

 

Note: if we could prove that the sup is infinite, this would easily confirm point (3) so the non-existant limit would ensue. If, otoh, lim(n ---> infinity) did exist, by point (1) it could only be zero, hence finite, and consequently R would have a finite max for some finite N but this doesn't strike me likely.

 

Still, R could well have some finite sup without this meaning that lim(n ---> infinity) exists.

[Turtle]

___Excellent elucidation! What-"sup' takes on a whole new meaning. :( Sanctus, thanks to you my fish wants a bike! :(

___Seriously though, I am honored to have sparked so much sharp discussion. You all continue to provide perspectives, terms, & information I lacked. May Hypography & you all prosper in direct proportion to Big R growing without bound.

___As it is, I wish to ask you all to consider the related threads "Distribution of Integers..." & "Strange Numbers..." & further employ your sharp minds in the pursuit of these topics. So consisely have you cornered Big R, that I hope to learn more from you concerning these other topics. I'm counting on you! :(

[Qfwfq]

It is easier to reckon on it using the sum which includes the numer itself as a divisor, I will define Q = R + 1. It isn't easy to spell out the method in detail without math notation, I'll just outline it.

 

Loosely, the upper bound of Q can be given as the product of the terms p/(p - 1) for all prime numbers p. More strictly this can be stated as the limit of a sequence, defined with a recursive rule, beginning with the values:

 

1, 2, 3, 15/4...

 

and each value given by the previous one times the above ratio, having the next prime as p. The last value above is hence given by:

 

1 * 2/(2 - 1) * 3/(3 - 1) * 5/(5 - 1) = 15/4

 

The recursive rule for R would be a bit less simple and, of course, it just gives values wich are decreased by 1. Whether the upper bound is finite or not isn't trivial to say yet. Perhaps we'd need to reckon on the asymptotic decrease in frequency of primes. Offhand, I can't remember for sure but I think the form is like (ln x)/x, I'll look it up.

 

The ratio between consecutive values approaches 1 from above for larger and larger primes, this is the type of sequence for which it isn't easy to distinguish if it is convergent or divergent. Using the differential increase 1/(p - 1) it can be regarded as a summation for which the ratio criterion doesn't decide between convergent and divergent. I'm beginning to think it could be convergent because of the decreasing frequency of primes, by asymptotic comparison with the summation of n raised to an exponent less than -1.

 

In any case, somebody may have worked out if the upper bound is finite or not, multiperfect numbers have been studied since Descartes, Fermat and Mersenne. I have emailed a friend who might know, or at least have an idea.

 

http://www-maths.swan.ac.uk/pgrads/bb/project/node27.html

[Turtle]

___I gave a similar, albeit more expansive, link in post#7 regarding pluperfect or k-fold numbers. Again, my Big R is not restricted to integral values, ie. pluperfect numbers only concern R when R is an integer.

___I question the use of the term "easier' in relation to whether one includes the number in the sum or not; it is better to simply state the two methods differ & how.

___In regard to the Big R of primes, it is decreasing at the same "rate" as the primes increase & so well studied; that is to say it doesn't interest me here.

___Finally, thank you for casting my work in the light of such luminaries as Descarte & Fermat. I only wish to fill in some of that which they missed.:cup:

[Qfwfq]

Perhaps my brief outline on Friday wasn't very clear, sorry. My analysis is not restricted to primes, it is based on them because they are fundamental to natural and integer numbers. The prime factorization of a general N can make calculations such as these more efficient. It certainly gives great possibilities in formal calculations.

 

Also, I was talking about the summation of terms 1/(p_n - 1), not the sequence of single terms. I gave this as the increment of the sequence but I let a factor slip out, it should have been R_(n-1)/(p_n - 1). It was the end of a long week of stress. Hang me.

 

This makes for the sequence being divergent and so is the summation I gave, according to my friend it is known to be so. Hence the right summation, with the increasing numerator, is also divergent. Sure enough, with a C program I ran on Saturday, I got a more or less log-type growth. It might be asymptotically logarithmic, base a bit more than 1.7, or it could be slightly less than logarithmic, I can't say for certain. It sure doesn't go toward a finite upper bound. By the results of this program, to get R values around 38 you need to multiply all primes up to a few billion, after raising the lower ones to high exponents. The resulting numers would be huge indeed and they would need to be huger and huger to get increasing R values.

 

In order to have R > 2.75 a number must necessarily be a multiple of 30. For R > 3.375 it must be a multiple of 210. For R > 3,7125 it must be a multiple of 2310. For R > 4.213542 it's gotta be multiple of 30030......

 

These conditions are necessary although not sufficient, they can be safely used to restrict an exhaustive search, such as you are doing.

As I questioned before, as integers grow, do you continue to have numbers abundant by less than 12?:eek:

I'm thinking about it and I'm not sure yet. I'll need to think after office hours.

[Turtle]

___Excellent work! No hanging! You interest, insight, information, & enthusiasm is great!

___You said Qfwfq, " Sure enough, with a C program I ran on Saturday, I got a more or less log-type growth. It might be asymptotically logarithmic, base a bit more than 1.7, or it could be slightly less than logarithmic." Wouldn't it be cool if the logaritmic base was the 'golden ratio' or 'e' or the square root of 3, or some other interesting number?

___Do not take my knocking of primes or expressions/formulae too seriously; the fact is they led me to this line of experiment. As a generalist however, I incline to examing as many specific cases as possible & so I turned to using the expressions to generate the verbose lists I habitually post.

___Keep up the good work & if you haven't read the related thread "Statistical Distribution..." I encourage you to do so.:eek:

[Qfwfq]

___Excellent work! No hanging!

Except that I don't work very well during spare moments here in the office, when I can post.

 

The filtering criteria for an exhaustive search are a bit less simple than I thought yesterday, an algorithm could be worked out though. Not easy to go into detail on these boards and without more spare time. I'll think about a clearer outline when I can.

[Turtle]

___Please take your time. I have worked on this since I first found Strange Numbers in 1996, & already thanks to those of you responding I have new perspectives on this work. I am honored that you find this topic worthy of your spare time & that you have shared it with others outside the board.:naughty:

[Turtle]

Ok then, I have new data. If you remember, we convinced ourselves there is no phattest number, & we turned our attention to thin numbers. I still believe the thinnest numbers, ie. the smallest Big R, belong to the largest Strange Numbers; however, I have a list here of 'thin Phat' numbers. The list is all the integers less than 1,000,000 that are abundant by less than 12. Here they are:

 

 

 

_Integer____Sum___Diff___#of-Divsrs__Big R___

______4_______5____1_________4_________1.25

_____12______16____4_________6_________1.33333333333333333333

_____16______19____3_________6_________1.1875

_____18______21____3_________6_________1.16666666666666666666

_____20______22____2_________6_________1.1

_____40______50___10_________8_________1.25

_____56______64____8_________8_________1.14285714285714285714

_____64______71____7_________8_________1.109375

_____70______74____4_________8_________1.05714285714285714285

_____88______92____4_________8_________1.04545454545454545454

____104_____106____2_________8_________1.01923076923076923076

____368_____376____8________10_________1.02173913043478260869

____464_____466____2________10_________1.00431034482758620689

____650_____652____2________12_________1.00307692307692307692

____836_____844____8________12_________1.00956937799043062200

___1696____1706___10________12_________1.00589622641509433962

___1888____1892____4________12_________1.00211864406779661016

___1952____1954____2________12_________1.00102459016393442622

___4030____4034____4________16_________1.00099255583126550868

___5830____5834____4________16_________1.00068610634648370497

___8925____8931____6________24_________1.00067226890756302521

__11096___11104____8________16_________1.00072098053352559480

__17816___17824____8________16_________1.00044903457566232599

__32128___32132____4________16_________1.00012450199203187250

__32445___32451____6________24_________1.00018492834026814609

__45356___45364____8________24_________1.00017638239703677572

__77744___77752____8________20_________1.00010290183165260341

__91388___91396____8________24_________1.00008753884536262966

_130304__130306____2________18_________1.00001534872298624754

_254012__254020____8________24_________1.00003149457505944601

_522752__522754____2________20_________1.00000382590597453476

[Turtle]

___Well now, I continue to find thin numbers abundant < 12; somewhere I postulated otherwise.

___I was rereading the entire thread & in post#10 Qwfwq layed out some interesting expressions in regard to Big R. Q, what do you think about these 'thin' numbers in the list in regard to your expression just referenced?

___The hunt goes on; tilting at windmills.:hyper:

 

_Integer____Sum___Diff___#of-Divsrs__Big R___

_______4________5____1________4_________1.25

______12_______16____4________6_________1.33333333333333333333

______16_______19____3________6_________1.1875

______18_______21____3________6_________1.16666666666666666666

______20_______22____2________6_________1.1

______40_______50___10________8_________1.25

______56_______64____8________8_________1.14285714285714285714

______64_______71____7________8_________1.109375

______70_______74____4________8_________1.05714285714285714285

______88_______92____4________8_________1.04545454545454545454

_____104______106____2________8_________1.01923076923076923076

_____368______376____8_______10_________1.02173913043478260869

_____464______466____2_______10_________1.00431034482758620689

_____650______652____2_______12_________1.00307692307692307692

_____836______844____8_______12_________1.00956937799043062200

____1696_____1706___10_______12_________1.00589622641509433962

____1888_____1892____4_______12_________1.00211864406779661016

____1952_____1954____2_______12_________1.00102459016393442622

____4030_____4034____4_______16_________1.00099255583126550868

____5830_____5834____4_______16_________1.00068610634648370497

____8925_____8931____6_______24_________1.00067226890756302521

___11096____11104____8_______16_________1.00072098053352559480

___17816____17824____8_______16_________1.00044903457566232599

___32128____32132____4_______16_________1.00012450199203187250

___32445____32451____6_______24_________1.00018492834026814609

___45356____45364____8_______24_________1.00017638239703677572

___77744____77752____8_______20_________1.00010290183165260341

___91388____91396____8_______24_________1.00008753884536262966

__130304___130306____2_______18_________1.00001534872298624754

__254012___254020____8_______24_________1.00003149457505944601

__522752___522754____2_______20_________1.00000382590597453476

_1848964__1848968____4_______36_________1.00000216337365140695

_2087936__2087944____8_______22_________1.00000383153506620892

_2291936__2291944____8_______24_________1.00000349049886209737

[Qfwfq]

Q, what do you think about these 'thin' numbers in the list in regard to your expression just referenced?

I found an interesting recipe, Turtle, try these numbers:

 

1) 8378368

2) 8382464

3) 33501184

4) 134193152

5) 1073119232

 

You should find them interesting, perhaps (1) is not as good but it might be interesting to compare it with (2). :Alien:

[Qfwfq]

Sorry, (5) won't be interesting, I have sharpened my focus on the recipe. Here it is:

 

1)Consider G = 2 raised to a power n

 

2)Find the highest prime p < 2G - 1

 

3)If 2G - p - 1 is less than 12, you've got one! Just multiply G*p.

 

If 2G - 1 is itself prime then it's a Mersenne prime and its product with G is a perfect number.

[Turtle]

___I have them! I'm off to have a closer look. My factoring run which generated the list I posted is now at 4 million plus & no new thin numbers. I better check that expression you gave too. You can count on me. :Alien:

[Turtle]

___I'm back! Nice job Q! First, your list of five numbers; you were correct #5 isn't interesting, or at least it's not 'thin'. #'s 1, 2 & 4, however are 'thin' & just what we want.

___The 3rd number in the list however...my friend that is an anomolous Strange Number! Remember a Strange Number is one which is abundant exactly by twelve, & further an anomolous Strange Number does not have a perfect number (6) as a factor. I never searched that high for them & the biggest I ever found was 127 thousand something(it's posted in the Strange Number thread) Of course, I stopped at 12 million & others may lie between 12 million & yours, but as of now you have officially found the largest anomolous Strange Number! Way to go!

___As to the recipe; it does produce thin numbers, but not all. When n=1, you get 4, when n=2 you get 20. In between 4 & 20 are the thin numbers 12, 16, & 18. Likewise when n=3 you get 104, but that passes over 40, 56, 64, 70, & 88 which are 'thin'

___Well, good progress. Onward & upward. :confused:

[Turtle]

___Maybe we can tweak the recipe so it finds others in the complete list? If we construct a suite of recipes, we can then make a cookbook for Thin Numbers!

 

___Double nice job by the way on your record find of the largest known anomolous Strange number. It won't be long I imagine before the numbers excede my capacity to check them. Yeah C1ay! Yeah Hypography! Yeah Strange Numbers, Thin Numbers, Big R & the gang!