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#154 modest

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Posted 22 February 2009 - 03:55 PM

for [math]n=56[/math], [math]2^{n-1}-13[/MATH] = [math]2^{55}-13[/math] = [math]36028797018963968-13[/MATH] = [math]36028797018963955[/MATH]. While the Numberator chokes on an integer this big, we can see it is divisible by 5 by the last digit. :clue:


I think you accidentally did [math]2^{n-1}-13[/math] instead of [math]2^{n+1}-13[/math].

I get [math]2^{n+1}-13[/math] = 144115188075855859

and [math]2^n(2^{n+1}-13)[/math] (possible strange anomaly) = 10384593717069654320312270165377024

144115188075855859 is prime, and 10384593717069654320312270165377024 is not divisible by 6. As for being abundant by 12... not sure yet...

~modest
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#155 TheBigDog

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Posted 22 February 2009 - 04:58 PM

We are starting to deal with some big numbers now hombres! I understand that I can inherit J# classes into vb.net, which include BigInt; the big as you need it to be number class. I will include that in how the 2K9 version works.

As you were then...

Bill
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#156 modest

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Posted 24 February 2009 - 04:30 PM

Here are the first few factors, T... along with the ongoing sum under each set...

5192296858534827160156135082688512
2
divisor sum = 5192296858534827160156135082688515

2596148429267413580078067541344256
4
divisor sum = 7788445287802240740234202624032775

1298074214633706790039033770672128
8
divisor sum = 9086519502435947530273236394704911

649037107316853395019516885336064
16
divisor sum = 9735556609752800925292753280040991

324518553658426697509758442668032
32
divisor sum = 10060075163411227622802511722709055

162259276829213348754879221334016
64
divisor sum = 10222334440240440971557390944043135

81129638414606674377439610667008
128
divisor sum = 10303464078655047645934830554710271

40564819207303337188719805333504
256
divisor sum = 10344028897862350983123550360044031

20282409603651668594359902666752
512
divisor sum = 10364311307466002651717910262711295

10141204801825834297179951333376
1024
divisor sum = 10374452512267828486015090214045695

5070602400912917148589975666688
2048
divisor sum = 10379523114668741403163680189714431

2535301200456458574294987833344
4096
divisor sum = 10382058415869197861737975177551871

1267650600228229287147493916672
8192
divisor sum = 10383326066469426091025122671476735

633825300114114643573746958336
16384
divisor sum = 10383959891769540205668696418451455

316912650057057321786873479168
32768
divisor sum = 10384276804419597262990483291963391

158456325028528660893436739584
65536
divisor sum = 10384435260744625791651376728768511

79228162514264330446718369792
131072
divisor sum = 10384514488907140055981823447269375

39614081257132165223359184896
262144
divisor sum = 10384554102988397188147046806716415

19807040628566082611679592448
524288
divisor sum = 10384573910029025754229658486833151

9903520314283041305839796224
1048576
divisor sum = 10384583813549340037270964327677951

4951760157141520652919898112
2097152
divisor sum = 10384588765309497178791617249673215

2475880078570760326459949056
4194304
divisor sum = 10384591241189575749551943713816575

1237940039285380163229974528
8388608
divisor sum = 10384592479129615034932106952179711

618970019642690081614987264
16777216
divisor sum = 10384593098099634677622188583944191

309485009821345040807493632
33554432
divisor sum = 10384593407584644498967229424992255

154742504910672520403746816
67108864
divisor sum = 10384593562327149409639749895847935

77371252455336260201873408
134217728
divisor sum = 10384593639698401864976010231939071

38685626227668130100936704
268435456
divisor sum = 10384593678384028092644140601311231

19342813113834065050468352
536870912
divisor sum = 10384593697726841206478206188650495

9671406556917032525234176
1073741824
divisor sum = 10384593707398247763395239787626495

4835703278458516262617088
2147483648
divisor sum = 10384593712233951041853758197727231

The sum already has the factor one in it, although it is not shown in the list. Are we sure that all the factors will be a power of 2?

~modest

#157 modest

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Posted 24 February 2009 - 04:46 PM

Are we sure that all the factors will be a power of 2?


If that's the case then I get this:

5192296858534827160156135082688512
2
divisor sum = 5192296858534827160156135082688515

2596148429267413580078067541344256
4
divisor sum = 7788445287802240740234202624032775

1298074214633706790039033770672128
8
divisor sum = 9086519502435947530273236394704911

649037107316853395019516885336064
16
divisor sum = 9735556609752800925292753280040991

324518553658426697509758442668032
32
divisor sum = 10060075163411227622802511722709055

162259276829213348754879221334016
64
divisor sum = 10222334440240440971557390944043135

81129638414606674377439610667008
128
divisor sum = 10303464078655047645934830554710271

40564819207303337188719805333504
256
divisor sum = 10344028897862350983123550360044031

20282409603651668594359902666752
512
divisor sum = 10364311307466002651717910262711295

10141204801825834297179951333376
1024
divisor sum = 10374452512267828486015090214045695

5070602400912917148589975666688
2048
divisor sum = 10379523114668741403163680189714431

2535301200456458574294987833344
4096
divisor sum = 10382058415869197861737975177551871

1267650600228229287147493916672
8192
divisor sum = 10383326066469426091025122671476735

633825300114114643573746958336
16384
divisor sum = 10383959891769540205668696418451455

316912650057057321786873479168
32768
divisor sum = 10384276804419597262990483291963391

158456325028528660893436739584
65536
divisor sum = 10384435260744625791651376728768511

79228162514264330446718369792
131072
divisor sum = 10384514488907140055981823447269375

39614081257132165223359184896
262144
divisor sum = 10384554102988397188147046806716415

19807040628566082611679592448
524288
divisor sum = 10384573910029025754229658486833151

9903520314283041305839796224
1048576
divisor sum = 10384583813549340037270964327677951

4951760157141520652919898112
2097152
divisor sum = 10384588765309497178791617249673215

2475880078570760326459949056
4194304
divisor sum = 10384591241189575749551943713816575

1237940039285380163229974528
8388608
divisor sum = 10384592479129615034932106952179711

618970019642690081614987264
16777216
divisor sum = 10384593098099634677622188583944191

309485009821345040807493632
33554432
divisor sum = 10384593407584644498967229424992255

154742504910672520403746816
67108864
divisor sum = 10384593562327149409639749895847935

77371252455336260201873408
134217728
divisor sum = 10384593639698401864976010231939071

38685626227668130100936704
268435456
divisor sum = 10384593678384028092644140601311231

19342813113834065050468352
536870912
divisor sum = 10384593697726841206478206188650495

9671406556917032525234176
1073741824
divisor sum = 10384593707398247763395239787626495

4835703278458516262617088
2147483648
divisor sum = 10384593712233951041853758197727231

2417851639229258131308544
4294967296
divisor sum = 10384593714651802681083020624003071

1208925819614629065654272
8589934592
divisor sum = 10384593715860728500697658279591935

604462909807314532827136
17179869184
divisor sum = 10384593716465191410504989992288255

302231454903657266413568
34359738368
divisor sum = 10384593716767422865408681618440191

151115727451828633206784
68719476736
divisor sum = 10384593716918538592860578971123711

75557863725914316603392
137438953472
divisor sum = 10384593716994096456586630726680575

37778931862957158301696
274877906944
divisor sum = 10384593717031875388449862762889215

18889465931478579150848
549755813888
divisor sum = 10384593717050764854381891097853951

9444732965739289575424
1099511627776
divisor sum = 10384593717060209587348729899057151

4722366482869644787712
2199023255552
divisor sum = 10384593717064931953833798567100415

2361183241434822393856
4398046511104
divisor sum = 10384593717067293137079631436005375

1180591620717411196928
8796093022208
divisor sum = 10384593717068473728709144940224511

590295810358705598464
17592186044416
divisor sum = 10384593717069064024537095831867391

295147905179352799232
35184372088832
divisor sum = 10384593717069359172477459556755455

147573952589676399616
70368744177664
divisor sum = 10384593717069506746500417977332735

73786976294838199808
140737488355328
divisor sum = 10384593717069580533617450303887871

36893488147419099904
281474976710656
divisor sum = 10384593717069617427387072699698431

18446744073709549952
562949953421312
divisor sum = 10384593717069635874694096362669695

144115188075855859
72057594037927936
divisor sum = 10384593717069636090866878476453490

The factors in my previous post were found exhaustively. These were found by checking powers of 2 with the prime at the end. If this is the whole list then the number is unfortunately not abundant by 12. :)

~modest

EDIT:

You can use this page as a calculator for big numbers:
http://gmplib.org/
at the bottom under Demo 1

#158 modest

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Posted 24 February 2009 - 05:37 PM

:confused: :doh: :doh:

Sorry, C is playing tricks on me. Here are all the factors...

5192296858534827160156135082688512
2
divisor sum = 5192296858534827160156135082688515

2596148429267413580078067541344256
4
divisor sum = 7788445287802240740234202624032775

1298074214633706790039033770672128
8
divisor sum = 9086519502435947530273236394704911

649037107316853395019516885336064
16
divisor sum = 9735556609752800925292753280040991

324518553658426697509758442668032
32
divisor sum = 10060075163411227622802511722709055

162259276829213348754879221334016
64
divisor sum = 10222334440240440971557390944043135

81129638414606674377439610667008
128
divisor sum = 10303464078655047645934830554710271

40564819207303337188719805333504
256
divisor sum = 10344028897862350983123550360044031

20282409603651668594359902666752
512
divisor sum = 10364311307466002651717910262711295

10141204801825834297179951333376
1024
divisor sum = 10374452512267828486015090214045695

5070602400912917148589975666688
2048
divisor sum = 10379523114668741403163680189714431

2535301200456458574294987833344
4096
divisor sum = 10382058415869197861737975177551871

1267650600228229287147493916672
8192
divisor sum = 10383326066469426091025122671476735

633825300114114643573746958336
16384
divisor sum = 10383959891769540205668696418451455

316912650057057321786873479168
32768
divisor sum = 10384276804419597262990483291963391

158456325028528660893436739584
65536
divisor sum = 10384435260744625791651376728768511

79228162514264330446718369792
131072
divisor sum = 10384514488907140055981823447269375

39614081257132165223359184896
262144
divisor sum = 10384554102988397188147046806716415

19807040628566082611679592448
524288
divisor sum = 10384573910029025754229658486833151

9903520314283041305839796224
1048576
divisor sum = 10384583813549340037270964327677951

4951760157141520652919898112
2097152
divisor sum = 10384588765309497178791617249673215

2475880078570760326459949056
4194304
divisor sum = 10384591241189575749551943713816575

1237940039285380163229974528
8388608
divisor sum = 10384592479129615034932106952179711

618970019642690081614987264
16777216
divisor sum = 10384593098099634677622188583944191

309485009821345040807493632
33554432
divisor sum = 10384593407584644498967229424992255

154742504910672520403746816
67108864
divisor sum = 10384593562327149409639749895847935

77371252455336260201873408
134217728
divisor sum = 10384593639698401864976010231939071

38685626227668130100936704
268435456
divisor sum = 10384593678384028092644140601311231

19342813113834065050468352
536870912
divisor sum = 10384593697726841206478206188650495

9671406556917032525234176
1073741824
divisor sum = 10384593707398247763395239787626495

4835703278458516262617088
2147483648
divisor sum = 10384593712233951041853758197727231

2417851639229258131308544
4294967296
divisor sum = 10384593714651802681083020624003071

1208925819614629065654272
8589934592
divisor sum = 10384593715860728500697658279591935

604462909807314532827136
17179869184
divisor sum = 10384593716465191410504989992288255

302231454903657266413568
34359738368
divisor sum = 10384593716767422865408681618440191

151115727451828633206784
68719476736
divisor sum = 10384593716918538592860578971123711

75557863725914316603392
137438953472
divisor sum = 10384593716994096456586630726680575

37778931862957158301696
274877906944
divisor sum = 10384593717031875388449862762889215

18889465931478579150848
549755813888
divisor sum = 10384593717050764854381891097853951

9444732965739289575424
1099511627776
divisor sum = 10384593717060209587348729899057151

4722366482869644787712
2199023255552
divisor sum = 10384593717064931953833798567100415

2361183241434822393856
4398046511104
divisor sum = 10384593717067293137079631436005375

1180591620717411196928
8796093022208
divisor sum = 10384593717068473728709144940224511

590295810358705598464
17592186044416
divisor sum = 10384593717069064024537095831867391

295147905179352799232
35184372088832
divisor sum = 10384593717069359172477459556755455

147573952589676399616
70368744177664
divisor sum = 10384593717069506746500417977332735

73786976294838199808
140737488355328
divisor sum = 10384593717069580533617450303887871

36893488147419099904
281474976710656
divisor sum = 10384593717069617427387072699698431

18446744073709549952
562949953421312
divisor sum = 10384593717069635874694096362669695

9223372036854774976
1125899906842624
divisor sum = 10384593717069645099192033124287295

4611686018427387488
2251799813685248
divisor sum = 10384593717069649713129851365360031

2305843009213693744
4503599627370496
divisor sum = 10384593717069652023476460206424271

1152921504606846872
9007199254740992
divisor sum = 10384593717069653185405164068012135

576460752303423436
18014398509481984
divisor sum = 10384593717069653779880314880917555

288230376151711718
36028797018963968
divisor sum = 10384593717069654104139488051593241

144115188075855859
72057594037927936
divisor sum = 10384593717069654320312270165377036

It is abundant by 12, it is a strange anomaly :)

~modest

#159 modest

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Posted 25 February 2009 - 01:02 AM

So Modest, do you understand what Bombdy did to get this? I'm still befuddled. :eek2:


Well... I'm not entirely sure. It looks like—if you write the factors of 304 as an example:

Posted Image

Noticing 19 is prime and 16 is a power of 2, this can be generalized such that,

Posted Image

The factors on the left can be summed,

[math]\sum_{i=0}^{n}2^i[/math]

and on the right (with the exception of 304 since you don't add the number itself as a factor,

[math]\sum_{i=1}^{n} 2^{n-i} \cdot p[/math]

When these numbers are summed they will need to be 12 greater than the number being factored. The number being factored can be written as 16 x 19 or [math]2^n \cdot p[/math], so this gives an equality,

[math]2^n \cdot p + 12=\sum_{i=0}^{n}2^i+\sum_{i=1}^{n}2^{n-i} \cdot p[/math]

Bombadil then rewrites the right hand side as,

[math]2^{n+1}-1+(2^{n}-1)p[/math]

making the equality,

[math]2^n \cdot p + 12 = 2^{n+1} -1 + (2^{n}-1) p[/math]

and rearranging,

[math]p=2^{n+1} - 13[/math]

It's quite ingenious if you ask me.

Before we get too drunk with euphoria, let's remember that this expression/equation Bomby gave us does not give us the Strange Anomaly 54 and so we can't assume there aren't other anomalies of that form.

Right. The factors of 54 don't follow the power of 2 format, yet it's still abundant by 12 and still can't be divided by 6. So, we can say any solution to Bombadil's method should give a strange anomaly, but others do exist.

~modest

#160 modest

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Posted 25 February 2009 - 02:43 PM

Back on the ground, I waited all night before posting to correct Modestino's last bit above, as I thought he'd be back & put in the fix. Do y'all see it? :daydreaming: :confused: Well, obviously 54 does divide by 6, as 6*9=54. 54 is a Strange Anomaly though, as 9 is not Prime.


:sherlock: 6 and non-prime :turtle:

I can't seem to keep my strangeness straight :piratesword:

~modest

#161 Bombadil

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Posted 28 February 2009 - 11:03 PM

Bombadilo, I would love to hear you talk about what exactly you saw that put you on this and other gory details from this result of the mathematical & insightful kind. :daydreaming:


I was just looking at your list of strange numbers wondering if it would be possible to generate more of them when I noticed that on all but one of them one of the numbers have a number of the form [math]2^n[/math] as one of the factors. After considering a few things that wouldn’t work I realized that the number could be factored out as [math]2^np[/math] where p was a prime number. Writing it like this makes it relatively easy to add up the sum of the factors as well as making the question of what value to give the prime a seemingly obvious question.

As for what to do next, I haven’t tried writing it out yet but I suspect that by choosing the form of a strange number as [math]2^npq[/math] with p and q being primes such that [math]p\neq q[/math] and [math]p,q>3[/math] and solving for one of them, that a contradiction may be reached but I haven’t had the chance to try it to find out for sure so maybe its just a dead end.

P.S. I see that it also can easily be used to generate strange numbers in other number sets as well so I wonder what other patterns might be found with it.

#162 Rade

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Posted 28 June 2010 - 09:18 PM

Most likely this is off topic, but here is an observation about this comment:

Take a pile of twelve beans. How many ways can you divide then into equal piles? One pile of twelve, two piles of six, three piles of four, four piles of three, & twelve piles of one.QED :shrug:

What about, 6 piles of 2 ? Do you not need the opposite of 2 piles of 6 ?

So, for number 12, you have a "strange number rule" of "opposites piles":

1 pile 12 and 12 piles 1
2 -- 6 and 6 -- 2 (add word pile(s) as needed)
3 -- 4 and 4 -- 3

(note: 5, 7, 9 never enter the rule)

==

OK, let us look at 12 * 2 = 24

1 -- 24 and 24 -- 1
2 -- 12 and 12 -- 2
3 -- 8 and 8 -- 3
4 -- 6 and 6 -- 4

(again 5,7,9 never enter the rule)

==

(this is fun)
24 * 2 = 48

1-- 48 and 48 -- 1
2 -- 24 and 24 -- 2
3 -- 16 and 16 -- 3
4 -- 12 and 12 -- 4
6 -- 8 and 8 -- 6

(again no 5, 7, 9 ), but I do see that you add 1 "opposite pile" each time you multiply by 2 )

48 * 2 = 96

1 -- 96 and 96 -- 1
2 -- 48 and 48 -- 2
3 -- 32 and 32 -- 3
4 -- 24 and 24 -- 4
6 -- 16 and 16 -- 6
8 -- 12 and 12 -- 8

(5, 7, 9 are no fun at all)
==

Let us look at 5, 7, 9 to see what is going on:

12 / 5 = 2.4000000000000000000.....(infinity, a repeating decimal 0)
24 / 5 = 4.8000000000000000000....
48 / 5 = 9.6000000000000000000....
96 / 5 = 19.200000000000000000....

(well, 5 likes infinity)

--
12 / 7 = 1.714285
24 / 7 = 3.4285714
48 / 7 = 6.8571428
96 / 7 = 13.714285

(I see a repeating pattern for 7)

--

12 / 9 = 1.33333333333333333333333....(to infinity)
24 / 9 = 2.66666666666666666666666.....
48 / 9 = 5.33333333333333333333333.....
96 / 9 = 10.6666666666666666666666....

(wow, a "repeating" repeating decimal for #9--first .33..then .66..then ..33..then..66)

Maybe 5, 7 9 are fun after all :idea:

==

OK, that was fun but I have absolutely no idea what it means, except I did not have to cut the grass.

#163 Gordon Freeman

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Posted 02 August 2010 - 03:19 AM

I ask the same question, why is twelve so special? I had heard about five being used within a special calculus formula in which sequences can be cracked like MD5 hash generators, but twelve makes no sense. And even if it is important, what can these numbers be used for? If it's just to simply make a mathematical code, then it can just as easily be cracked.

In fact, the only mathematical coding that has proven to be uncrackable thus far is nonfigurative sets. Special numbers can be easily solved for, so again, what is their significance?

#164 Gordon Freeman

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Posted 04 August 2010 - 01:54 AM

My only resources are my colleagues and my notebook that I've written some sequences into. I have a friend that uses this site that could help you much more greatly than I; his user-name is IDMclean. He knows quite a bit more about non-figuratives than I do.

If you need anything on physics, gravito force theory, therma-fluids, or time thread theory, I'd be glad to talk some more.
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#165 Rade

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Posted 24 August 2010 - 06:43 AM

hello turtle,

Not sure if you have noticed this, but there is a "perfect" (r^2 = 1) linear relationship when you log transform both sets of numbers (x,y plot) in the sequence below. You will see that all of your "numbers" (12-strange, 56-bizarre, 992-peculiar, 16256-curious) are included in the set of y numbers, plus a few others.

x y
4 -- 12
8 -- 56
16 - 240
32 - 992
64 - 4032
128 - 16256