Turtle Posted February 26, 2005 Author Report Posted February 26, 2005 ___Find attached below, lists of the integers classed by number of divisors. In the post above I attached the list of the class of numbers with 8 divisors. Here I add the classes 6 Divisors, 10 Divisors, & 12 Divisors. Each list is ordered.___We have discussed whether or not to include a number itself in the sum, & here I do not. I point out again, that the choice is arbitrary. Just so, the column labled "Sigma" is the sum of the proper divisors not including the number itself. The first column is the integer itself, the column labled 'Difference' is 'Sigma Minus Integer', & the colums labled 'Divisors' is the verbose list of proper divisors.___I further point out that if the Difference is 0 (zero), the number is Perfect, if the Difference is positive the number is abundant, & if the Difference is negative the number is derficient. Quote
Turtle Posted April 16, 2005 Author Report Posted April 16, 2005 ___I noticed Thelonius asked in a new thread about Prime Density, and so I thought to bring this topic back for review. What I describe here is not only Prime Density, but the Density of every class of numbers by their number of divisors.___As to what use it is, (beyond cryptography) we must further investigate in order to know. What is new & unique here is that beyond Prime & Pseudo-Prime Density, I have described a road map for extending the investigation beyond what others have. Quote
Turtle Posted May 10, 2005 Author Report Posted May 10, 2005 ___I have packed away my work on this topic; however it occured to me that since the density of primes study has a fair amount of data, & since the classes I outline here, ie. classifying all integers by their number of divisors, must use the primes in their members, the comparison of prime density/class density may have some interesting pattern. Quote
Turtle Posted May 13, 2005 Author Report Posted May 13, 2005 ___I noticed several Google spyders on this thread & the Strange Numbers thread as well several times & I encourage anyone furthering this work or questioning it to post.___One particular aspect of the different statistical distributions here I may have noted early on, has occupied my mind lately. As we invoked prime density & its relation & inclusion in the larger set of distributions by class of number of divisors, we implied but did not explicitly state that this density is steadily decreasing.___This is not the case for all the classes however, & in fact some class densitys appear to oscillate up & down in a manner similar to an attractor. There is much exploration here required beyond my computer capabilities & I invite anyone's participation. :friday: Quote
Turtle Posted June 8, 2005 Author Report Posted June 8, 2005 ___I have the promised data sheet now in the Hypography Science Gallery:http://hypography.com/gallery/showimage.php?i=283 ___I may have said it went through 13 million, however on finding the sheet I found it through 10 million. No matter. If you look at the right hand column of the sheet, you see arrows, some pointing up, some down, & some both. As I have discussed in this thread, those wavering up & down have the quality of Strange Attractors; on the sheet these attractors occur for the classes 6 pairs of divisors, 7 pairs, & 27 pairs.___Now on further examination, I see some blanks in the arrow column, but on the whole, only a few densities steadily increase over the range of the data, eg. class of 15 pairs, & class of 23 pairs.___The experiment continues then, mistakes & all. :hyper: Quote
CraigD Posted July 10, 2005 Report Posted July 10, 2005 … how weird is it that in the first 15 million+ integers, only 1 has 92 divisors, none have 94 divisors, & then all of a sudden 110,229 have 96 divisors!One of the interesting questions raised these brute-force generated tables has the form “for the first X integers, why do none have Y divisors?” Fortunately, some non-computer supported reasoning can answer this one.An integer N is uniquely defined by its prime factorization, that is, N =Product(Prime(i)^A(i)),where Prime(i) is the i-th prime, and A(i) is its exponent in the prime factorization of N. E.g: for N=20, A={2,0,1}, 28= 2^2 * 3^0 * 5^1The number of integer divisior of N,Divisors(N) = Product(A(i)+1),where A is as above. E.g: Divisors(20) = (2+1)*(0+1)*(1+1) = 6, 20={1*20, 2*10, 4*5, 5*4, 10*2, 20*1}. This can be applied to calculate the minimum integer that has any given number of divisors. Here’s the calculation for Divisors 1-12Divs / A / Min N1 / / 12 / 1 / 23 / 2 / 44 / 1,1 / 65 / 4 / 166 / 2,1 / 127 / 6 / 648 / 1,1,1 / 309 / 2,2 / 3610 / 4,1 / 4811 / 10 / 102412 / 2,1,1 / 60 So, for the first 100 integers, none will have 7 or 11 divisors, but some will have 12 divisors. For 46-48 divisors, we haveDivs / A / Min N46 / 22,1 / 12,582,91247 / 46 / 70,368,744,177,66448 / 2,1,1,1,1 / 4,620 This agrees with the observation that, for the first 15,000,000 integers, there is only 1 with 46 divisors, none with 47, but many with 48. Quote
Turtle Posted July 11, 2005 Author Report Posted July 11, 2005 ___How beautiful is that! Free at last! Or am I? You say "one of the intersting questions raised". That anyone finds it interesting at all makes all the brute force worth it. Thanks CraigD for you insighful & succinct analysis! Now where did I put those Strange Numbers? :) ___I see a discrepency in this list: Divs / A / Min N1 / / 12 / 1 / 23 / 2 / 44 / 1,1 / 65 / 4 / 166 / 2,1 / 127 / 6 / 648 / 1,1,1 / 309 / 2,2 / 3610 / 4,1 / 4811 / 10 / 102412 / 2,1,1 / 60___24 is the least integer with 8 divisors; 8 / 3,1 / 24 ? This may now take us back over to Strange Numbers as 24 & 30 are the 1st & 2nd elements in the set of Strange Numbers. Somewhere we noticed something about 54 (maybe the Katabatak thread), anyway 54 is something of a strange Strange number as it has the perfect 6 paired with a composite (9) rather than the usual perfect 6 & a prime. Quote
CraigD Posted July 14, 2005 Report Posted July 14, 2005 I see a discrepency in this list …Divs / A / Min N …8 / 1,1,1 / 30 …___24 is the least integer with 8 divisors; 8 / 3,1 / 24 …You’re correct. 30 has 8 Divs, but isn’t the smallest integer that does. I generated the short table in my 7/10/05 post by hand, using (faulty!) intuition to find the A that gives the smallest N. Since then, I’ve written a program to generate the list. Though it’s slightly complicated, it’s non-itterative – that is, it finds the A for the smallest N other than by exhaustively trying every possibility.M code:k r X(1),!,X(2),!,X(3),!,X(4),!,X(5),!,X(6) s P=2,P(1)=2,P(2)=3 x X(1)f n=1:1 x X(2),X(4) w n," " x X(6) w " ",m,! ;-1n (X,A,n,P) k A s q=n f i=1:1 q:q=1 x:'$d(P(i)) X(3) f q:q#P(i) s A(i)=$g(A(i))+1,q=qP(i) ;factor n into A(), grow P() -2n (P) f i=P(P)+2:2 x "f j=2:1 s:j*j>i j=0 q:'j q:i#j=0" i 'j s P=P+1,P(P)=i q ;grow P() -3n (X,B,A,P) k B s j="" f s j=$o(A(j),-1) q:'j s y=P(j) f i=1:1:A(j) x X(5) s B(k)=$g(B(k))+1*y-1 ;find B() given A() ;-4n (X,k,B,y,P) s m=0 f i=1:1 x X(3):'$d(P(i)),X(3):'$d(P(i+1)) s n=P(i)**($g(B(i))+1*(y-1)) s:n<m!'m m=n,k=i q:P(i+1)**(y-1)'<m ;-5n (B,P,m) s m=1,d="" f i=1:1:$o(B(""),-1) s m=P(i)**B(i)*m w d,B(i) s d="," ;expand factors B() into m -6 Since this is using the interpreter’s intrinsic math functions, it’s only able to generate up to Divs 306, and displays some imprecise Ns for large values. Output is at the end of this post. Having answered the “why are there gaps” question, I’m less interested now in this line of inquiry. The most beautiful question that I think can come out of the “Statisitical View” investigation is a generalized version of the one prompting the Prime Number Theorem – “How many prime numbers less than N are there?”. The famous answer – Pi(N) ~ N/Log(N) goes back a couple of centuries, has some famous proofs, and improvements. A more general question, basically the one this thread is asking, is “How many numbers less than N with X factors are there. I suspect that a function related to and as elegant as the PNT answers this question. PS: Min N for number of divisors 1-306 (some large N’s approximate)Divs / A / Min N1 / / 12 / 1 / 23 / 2 / 44 / 1,1 / 65 / 4 / 166 / 2,1 / 127 / 6 / 648 / 3,1 / 249 / 2,2 / 3610 / 4,1 / 4811 / 10 / 102412 / 2,1,1 / 6013 / 12 / 409614 / 6,1 / 19215 / 4,2 / 14416 / 3,1,1 / 12017 / 16 / 6553618 / 2,2,1 / 18019 / 18 / 26214420 / 4,1,1 / 24021 / 6,2 / 57622 / 10,1 / 307223 / 22 / 419430424 / 2,1,1,1 / 42025 / 4,4 / 129626 / 12,1 / 1228827 / 2,2,2 / 90028 / 6,1,1 / 96029 / 28 / 26843545630 / 4,2,1 / 72031 / 30 / 107374182432 / 3,1,1,1 / 84033 / 10,2 / 921634 / 16,1 / 19660835 / 6,4 / 518436 / 2,2,1,1 / 126037 / 36 / 6871947673638 / 18,1 / 78643239 / 12,2 / 3686440 / 4,1,1,1 / 168041 / 40 / 109951162777642 / 6,2,1 / 288043 / 42 / 439804651110444 / 10,1,1 / 1536045 / 4,2,2 / 360046 / 22,1 / 1258291247 / 46 / 7036874417766448 / 5,1,1,1 / 336049 / 6,6 / 4665650 / 4,4,1 / 648051 / 16,2 / 58982452 / 12,1,1 / 6144053 / 52 / 450359962737049654 / 2,2,2,1 / 630055 / 10,4 / 8294456 / 6,1,1,1 / 672057 / 18,2 / 235929658 / 28,1 / 80530636859 / 58 / 28823037615171174460 / 4,2,1,1 / 504061 / 60 / 115292150460684697662 / 30,1 / 322122547263 / 6,2,2 / 1440064 / 3,3,1,1 / 756065 / 12,4 / 33177666 / 10,2,1 / 4608067 / 66 / 7378697629483820648068 / 16,1,1 / 98304069 / 22,2 / 3774873670 / 6,4,1 / 2592071 / 70 / 118059162071741130400072 / 5,2,1,1 / 1008073 / 72 / 472236648286964521600074 / 36,1 / 20615843020875 / 4,4,2 / 3240076 / 18,1,1 / 393216077 / 10,6 / 74649678 / 12,2,1 / 18432079 / 78 / 30223145490365729380000080 / 4,3,1,1 / 1512081 / 2,2,2,2 / 4410082 / 40,1 / 329853488332883 / 82 / 483570327845851670000000084 / 6,2,1,1 / 2016085 / 16,4 / 530841686 / 42,1 / 1319413953331287 / 28,2 / 241591910488 / 10,1,1,1 / 10752089 / 88 / 30948500982134506880000000090 / 4,2,2,1 / 2520091 / 12,6 / 298598492 / 22,1,1 / 6291456093 / 30,2 / 966367641694 / 46,1 / 21110623253299295 / 18,4 / 2123366496 / 5,3,1,1 / 3024097 / 96 / 7922816251426433760000000000098 / 6,6,1 / 23328099 / 10,2,2 / 230400100 / 4,4,1,1 / 45360101 / 100 / 1267650600228229402000000000000102 / 16,2,1 / 2949120103 / 102 / 5070602400912917608000000000000104 / 12,1,1,1 / 430080105 / 6,4,2 / 129600106 / 52,1 / 13510798882111488107 / 106 / 81129638414606681760000000000000108 / 5,2,2,1 / 50400109 / 108 / 324518553658426727000000000000000110 / 10,4,1 / 414720111 / 36,2 / 618475290624112 / 6,3,1,1 / 60480113 / 112 / 5192296858534827632000000000000000114 / 18,2,1 / 11796480115 / 22,4 / 339738624116 / 28,1,1 / 4026531840117 / 12,2,2 / 921600118 / 58,1 / 864691128455135232119 / 16,6 / 47775744120 / 4,2,1,1,1 / 55440121 / 10,10 / 60466176122 / 60,1 / 3458764513820540928123 / 40,2 / 9895604649984124 / 30,1,1 / 16106127360125 / 4,4,4 / 810000126 / 6,2,2,1 / 100800127 / 126 / 85070591730234615920000000000000000000128 / 3,3,1,1,1 / 83160129 / 42,2 / 39582418599936130 / 12,4,1 / 1658880131 / 130 / 1361129467683753854000000000000000000000132 / 10,2,1,1 / 322560133 / 18,6 / 191102976134 / 66,1 / 221360928884514619400135 / 4,2,2,2 / 176400136 / 16,1,1,1 / 6881280137 / 136 / 87112285931760246640000000000000000000000138 / 22,2,1 / 188743680139 / 138 / 348449143727040986600000000000000000000000140 / 6,4,1,1 / 181440141 / 46,2 / 633318697598976142 / 70,1 / 3541774862152233912000143 / 12,10 / 241864704144 / 5,2,1,1,1 / 110880145 / 28,4 / 21743271936146 / 72,1 / 14167099448608935650000147 / 6,6,2 / 1166400148 / 36,1,1 / 1030792151040149 / 148 / 356811923176489970200000000000000000000000000150 / 4,4,2,1 / 226800151 / 150 / 1427247692705959881000000000000000000000000000152 / 18,1,1,1 / 27525120153 / 16,2,2 / 14745600154 / 10,6,1 / 3732480155 / 30,4 / 86973087744156 / 12,2,1,1 / 1290240157 / 156 / 91343852333181432400000000000000000000000000000158 / 78,1 / 906694364710971881400000159 / 52,2 / 40532396646334464160 / 4,3,1,1,1 / 166320161 / 22,6 / 3057647616162 / 5,2,2,2 / 352800163 / 162 / 5846006549323611672000000000000000000000000000000164 / 40,1,1 / 16492674416640165 / 10,4,2 / 2073600166 / 82,1 / 14507109835375550100000000167 / 166 / 93536104789177786700000000000000000000000000000000168 / 6,2,1,1,1 / 221760169 / 12,12 / 2176782336170 / 16,4,1 / 26542080171 / 18,2,2 / 58982400172 / 42,1,1 / 65970697666560173 / 172 / 5986310706507378348000000000000000000000000000000000174 / 28,2,1 / 12079595520175 / 6,4,4 / 3240000176 / 10,3,1,1 / 967680177 / 58,2 / 2594073385365405696178 / 88,1 / 928455029464035206000000000179 / 178 / 383123885216472214400000000000000000000000000000000000180 / 4,2,2,1,1 / 277200181 / 180 / 1532495540865888858000000000000000000000000000000000000182 / 12,6,1 / 14929920183 / 60,2 / 10376293541461622780184 / 22,1,1,1 / 440401920185 / 36,4 / 5566277615616186 / 30,2,1 / 48318382080187 / 16,10 / 3869835264188 / 46,1,1 / 1055531162664960189 / 6,2,2,2 / 705600190 / 18,4,1 / 106168320191 / 190 / 1569275433846670190000000000000000000000000000000000000000192 / 5,3,1,1,1 / 332640193 / 192 / 6277101735386680760000000000000000000000000000000000000000194 / 96,1 / 237684487542793012800000000000195 / 12,4,2 / 8294400196 / 6,6,1,1 / 1632960197 / 196 / 100433627766186892200000000000000000000000000000000000000000198 / 10,2,2,1 / 1612800199 / 198 / 401734511064747568800000000000000000000000000000000000000000200 / 4,4,1,1,1 / 498960201 / 66,2 / 664082786653543858300202 / 100,1 / 3802951800684688206000000000000203 / 28,6 / 195689447424204 / 16,2,1,1 / 20643840205 / 40,4 / 89060441849856206 / 102,1 / 15211807202738752820000000000000207 / 22,2,2 / 943718400208 / 12,3,1,1 / 3870720209 / 18,10 / 15479341056210 / 6,4,2,1 / 907200211 / 210 / 1645504557321206042000000000000000000000000000000000000000000000212 / 52,1,1 / 67553994410557440213 / 70,2 / 10625324586456701740000214 / 106,1 / 243388915243820045300000000000000215 / 42,4 / 356241767399424216 / 5,2,2,1,1 / 554400217 / 30,6 / 782757789696218 / 108,1 / 973555660975280181000000000000000219 / 72,2 / 42501298345826806940000220 / 10,4,1,1 / 2903040221 / 16,12 / 34828517376222 / 36,2,1 / 3092376453120223 / 222 / 6739986666787659948000000000000000000000000000000000000000000000000224 / 6,3,1,1,1 / 665280225 / 4,4,2,2 / 1587600226 / 112,1 / 15576890575604482900000000000000000227 / 226 / 107839786668602559200000000000000000000000000000000000000000000000000228 / 18,2,1,1 / 82575360229 / 228 / 431359146674410236800000000000000000000000000000000000000000000000000230 / 22,4,1 / 1698693120231 / 10,6,2 / 18662400232 / 28,1,1,1 / 28185722880233 / 232 / 6901746346790563788000000000000000000000000000000000000000000000000000234 / 12,2,2,1 / 6451200235 / 46,4 / 5699868278390784236 / 58,1,1 / 4323455642275676160237 / 78,2 / 2720083094132915644000000238 / 16,6,1 / 238878720239 / 238 / 441711766194596082400000000000000000000000000000000000000000000000000000240 / 4,2,1,1,1,1 / 720720241 / 240 / 1766847064778384330000000000000000000000000000000000000000000000000000000242 / 10,10,1 / 302330880243 / 8,2,2,2 / 2822400244 / 60,1,1 / 17293822569102704640245 / 6,6,4 / 29160000246 / 40,2,1 / 49478023249920247 / 18,12 / 139314069504248 / 30,1,1,1 / 112742891520249 / 82,2 / 43521329506126650300000000250 / 4,4,4,1 / 5670000251 / 250 / 1809251394333065554000000000000000000000000000000000000000000000000000000000252 / 6,2,2,1,1 / 1108800253 / 22,10 / 247669456896254 / 126,1 / 255211775190703847800000000000000000000255 / 16,4,2 / 132710400256 / 3,3,1,1,1,1 / 1081080257 / 256 / 115792089237316195400000000000000000000000000000000000000000000000000000000000258 / 42,2,1 / 197912092999680259 / 36,6 / 50096498540544260 / 12,4,1,1 / 11612160261 / 28,2,2 / 60397977600262 / 130,1 / 4083388403051261562000000000000000000000263 / 262 / 7410693711188236504000000000000000000000000000000000000000000000000000000000000264 / 10,2,1,1,1 / 3548160265 / 52,4 / 364791569817010176266 / 18,6,1 / 955514880267 / 88,2 / 2785365088392105619000000000268 / 66,1,1 / 1106804644422573097000269 / 268 / 474284397516047136400000000000000000000000000000000000000000000000000000000000000270 / 4,2,2,2,1 / 1940400271 / 270 / 1897137590064188546000000000000000000000000000000000000000000000000000000000000000272 / 16,3,1,1 / 61931520273 / 12,6,2 / 74649600274 / 136,1 / 261336857795280739900000000000000000000000275 / 10,4,4 / 51840000276 / 22,2,1,1 / 1321205760277 / 276 / 121416805764108067000000000000000000000000000000000000000000000000000000000000000000278 / 138,1 / 1045347431181122960000000000000000000000000279 / 30,2,2 / 241591910400280 / 6,4,1,1,1 / 1995840281 / 280 / 1942668892225729072000000000000000000000000000000000000000000000000000000000000000000282 / 46,2,1 / 3166593487994880283 / 282 / 7770675568902916288000000000000000000000000000000000000000000000000000000000000000000284 / 70,1,1 / 17708874310761169560000285 / 18,4,2 / 530841600286 / 12,10,1 / 1209323520287 / 40,6 / 801543976648704288 / 5,2,1,1,1,1 / 1441440289 / 16,16 / 2821109907456290 / 28,4,1 / 108716359680291 / 96,2 / 713053462628379038400000000000292 / 72,1,1 / 70835497243044678250000293 / 292 / 7957171782556586280000000000000000000000000000000000000000000000000000000000000000000000294 / 6,6,2,1 / 8164800295 / 58,4 / 23346660468288651260296 / 36,1,1,1 / 7215545057280297 / 10,2,2,2 / 11289600298 / 148,1 / 1070435769529469911000000000000000000000000000299 / 22,12 / 2229025112064300 / 4,4,2,1,1 / 2494800301 / 42,6 / 3206175906594816302 / 150,1 / 4281743078117879643000000000000000000000000000303 / 100,2 / 11408855402054064620000000000000304 / 18,3,1,1 / 247726080305 / 60,4 / 93386641873154605100306 / 16,2,2,1 / 103219200 Quote
Turtle Posted July 14, 2005 Author Report Posted July 14, 2005 )))Ooohhhh...I love a verbose list! ___I agree that with the mystery lightened, my interest wains as well. After the first short list, I was so satisfied that I threw out 30 pages of notes I've kept for qite some years. :eek: I did retrieve them when I found the discrepency in review; maybe now I'll just toss half out in view of the new development. :eek: ___Allowing this thread to langusih now, it's on to the Strange Numbers to try & put them to rest as well. Great work & participation CraigD; Thanks. :eek: Quote
Turtle Posted January 27, 2009 Author Report Posted January 27, 2009 :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: Quote
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