Big "R" & The Hunt For Phat Numbers

[Qfwfq]

The third: you're right! I hadn't noticed that its 2G - p - 1 is exactly 12, but it is! The recipe I found is good for two of the numbers that I checked yesterday from your list in #30:

 

130304 = 2^8 509

 

522752 = 2^9 1021

 

but not good for the other two I had checked:

 

254012 = 2^2 11 23 251

 

1848964 = 2^2 13 31^2 37

 

This shows the recipe is not exhaustive, indeed it coudn't be. One can always do some tinkering with prime factors, and powers of them, to find slightly abundant numbers. In any case, it helps to start with powers of two (see 254012 & 1848964). It would be somewhat awkward to spell out all the details, on these boards, but I can see now how it works.

 

Previously I had tried reasoning on variants of the recipe (6 times a prime) for non-anomalous strange numbers hoping to find something for thin numbers but I soon found this was off-track; I showed that such methods could not give an excess of less than 12. I stopped thinking about it, though I had in mind the idea of taking slightly deficient numbers and multiplying them by a suiteable prime. I didn't get the matter sorted out, despite knowing that powers of 2 easily give you a slighlty deficient number, and with a deficiency as slight as you like. Laziness. I was a hair's breadth from yesterday's recipe.

 

Then you posted your new lists and I factored the most interesting specimens to maybe find ideas and, seeing the cases of 130304 and 522752, it hit me... :confused:

 

___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!

I was hoping along these lines yesterday but, when I had a look at it after office hours, I realised something I hadn't considered. In any case, one can always do some tinkering to find slight deficiencies and slight abundancies and a suite of recipies wouldn't be exhaustive.

 

The only tweaking on my recipe, that I can see now, is to take a power of 2 and multiply it by more than one prime, having a product less than 2^n+1. To make it an exact recipe I would have to check it a bit. This would merge into pure tinkering. I think one could also find thin numbers that are products of large primes, perhaps with the largest PF being compareable to the product of the others, but not necessarily... I'd need to work it out a bit. :Alien: :hihi:

[Turtle]

___My factoring run is chunking along at about a million integers per day. It should be interesting to see if any more Thin numbers show up before I reach the first number in your list.

___Here's an interesting point, I think from one of the "Joy of Mathematics" books: All multiples of perfect numbers are abundant & all divisors of perfect numbers are deficient.

___I think I'll group numbers from the Thin list by class (ie. all abundant by 2 together etc.) to see if they have any commonalities similar to Strange. Common differences, factors etc.

___To the Hunt! :confused:

[Turtle]

___I am winding down my computer experimenting as it costs 15-20 cents a day to run. I intend to run a couple more days, as I am at just over 6 million integers seived & I want to fill the gap up to Q's recipe numbers 1 & 2 from seven posts ago. ( I just noticed that while you're composing a reply, the review view does not show a label for post #'s)

___Anyway, I'll still watch for questions or new discoveries from those of you now experimenting on your own, & report whether I find any new Thin numbers in the next couple of days. Good hunting! :)

[Turtle]

___Last night my experiment run reached Qwfwq's recipe Thin numbers. No Thin numbers lie between Q's & my last found. This concludes for the time being my active search. Below find the complete list of the only numbers abundant by less than 12 between 1 & 8,382,464, as well as their Big R, which is small, which is why these numbers are Thin (Thanks to memeber Lazlo Toth for the term 'Thin')

 

_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

_8378368__8378372____4_______24_________1.00000047741994622341

_8382464__8382466____2_______24_________1.00000023859333007573

 

 

:cup: :cup: :cup: :cup: :cup: :cup: :cup: :cup: :cup: :cup: :cup: :cup: :cup:

[Turtle]

___Just a note to sum things up. If the recipe for Thin Numbers Qwfwq put forward holds for infinite ingredients, ie. input grows without bound, then my conjecture that the largest Strange Numbers are the Thinnest fails. No doubt I will take up the subject again in the future. ;)

[Qfwfq]

The recipe isn't guaranteed to give output for larger and larger powers of two, as it depends on there being primes just less than a power of two (between 1 less and 1+12 less: 1 less means a Mersenne prime which gives a perfect number, 1+12 less gives a strange number). Despite this I think it likely that Thin and Strange numbers have no limit in size, in analogy to this being true for perfect numbers. I don't know the proof of it for perfect numbers but it might be possible to use a similar argument for the other two types.

 

Reasoning on the recipe and the possible variants and on the distribution of primes, it would appear that all three types of number should become more and more rare at increasing ranges of size, but with perfect and strange having comparable rarity and thin being less rare.

[Turtle]

Qwfwq said, "Reasoning on the recipe and the possible variants and on the distribution of primes, it would appear that all three types of number should become more and more rare at increasing ranges of size, but with perfect and strange having comparable rarity and thin being less rare."

___I have given this some thought and perfect & strange are not comparable, at least if by that you mean equally dense; remember some Strange Numbers (anomolous strange) have no perfect factors so they are more dense than perfect.

___I am not actively hunting Phat numbers, but I keep my gun well oiled for the next outing.

[Qfwfq]

This weekend, looking through a few printouts of some time ago for the one relevant to Cindy's identity, I took a glance at:

 

http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf

 

and I realized the first equation proves the upper bound on Big "R" is infinite!!!

 

The very first equation, the identity found by Euler, shows the case. The lhs can easily be manipulated so that, for the choice s = 1, it is the infinite product I had found as being the upper bound we had sought. My factors are just p/(p - 1). I got this infinite product from a way of calculating Q, and hence R by subtracting 1, given a number's prime factorization. The rhs of Euler's identity, for the same choice of s, is the harmonic series, well known to be divergent.

 

Hence the product of p/(p - 1) extended to all primes p is infinite and so is the upper bound on R, QED.

[Turtle]

___That is fantastic Q! I simply love Euler for his work, both volume & content. No small nod to you either for your output & content! I may have persisted in my computer runs for years with no idea what I was up against in the hunt for Big R & Phat numbers.

___I read the paper, but as you know my calculus is no calculus at all, & I barely recognized any of the argument. Again, nice job Q!

[Turtle]

___Just when all seemed resolved to me here, I realized there is still a little work to do. The last few posts resolved the infinite upper bound of the integer component of Big R, but not the fractional part. So what is the upper bound of the fractional part? It seems it must lie below .9999... algebraically, but how does it really behave? What is the real limit I mean, as opposed to a theoretical limit.

___Good enough reason to continue the search experiment as I outlined in the beginning of this thread, & alter the focus to just the fraction part of Big R. ;)

[Turtle]

___Now I haven't yet mentioned it, but there is the matter of Katabataks in all this Phatness. If you have no familiarity with Katabtaks I refer you to the thread by that name here at Hypography.

 

 

 

___Now that you have read up on Katabataks, notice the K Pattern for the multiperfect/pluperfect numbers.

___Just never something to not explore in the case of Phat Numbers. :eek: :hyper:

[Qfwfq]

So what is the upper bound of the fractional part? It seems it must lie below .9999... algebraically, but how does it really behave?

Actually, I'd say it could be arbitrarily near to 1. R values are certainly rational, and so the fractional part, but I see no reason for there to be an upper bound on the fractional part. Indeed, consider powers of 2: for n-->infinity the R value of 2^n tends to 1 from below.

 

How are the K patterns?

[Turtle]

___I had in mind Q the often referenced algebraic demonstration that the repeating decimal .999999 is equal to 1. :hyper:

___In regard to the K patterns I thought out loud there & haven't checked the multiperfect/pluperfect K Patterns. I have however regularly watched & noted the K Patterns of the Strange, Bizarre, Peculiar, etc Numbers, but chose to not include that data with my lists in the thread Strange Numbers for clarity's sake. :eek:

[Turtle]

:gift: Inasmuch as the topic of Strange Numbers has gained some new interest and activity in that thread, I decided to bump this thread to put the Strange Numbers in the context in which I found themI found them. Enjoy. :) :hihi: :friday: