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You are not logged in. #1 20060521 18:57:32
Cool Perfect Power Function!!!int x is a perfect power iff there exists int a and int b for which For example, 4 and 8 are perfect powers, but 10 isn't. The function gives the nth perfect square. etc. The sequence is very interesting. I'll post some properties of it. IPBLE: Increasing Performance By Lowering Expectations. #2 20060521 19:04:13
Re: Cool Perfect Power Function!!!First, a program. (C++ ) Code:#include<iostream> #include<fstream> #include<math.h> using namespace std; const long int n=1000000; int main(){ //making the array cout<<"n = "<<n<<endl; bool result[n]; for(long int i=1;i<n;i++){ result[i]=false; } for(long int i=2;i<=sqrt((double)n)+1;i++){ if(!result[i1]){ long int pw=i*i; while(pw<=n){ result[pw1]=true; pw*=i; } } } ofstream file("results.txt",ios::out); for(long int i=0;i<n;i++){ if(result[i]){ file<<i+1<<","; } } cout<<"For results, see 'results.txt'. Press any key to quit:"; char c; cin>>c; return c; } This is some kind of Erastrotenes prime sieve. IPBLE: Increasing Performance By Lowering Expectations. #3 20060521 19:06:34
Re: Cool Perfect Power Function!!!Here's S up to i=1000000(milion): Code:1,4,8,9,16,25,27,32,36,49,64,81,100,121,125,128,144,169, 196,216,225,243,256,289,324,343,361,400,441,484,512,529, 576,625,676,729,784,841,900,961,1000,1024,1089,1156,1225, 1296,1331,1369,1444,1521,1600,1681,1728,1764,1849,1936, 2025,2048,2116,2187,2197,2209,2304,2401,2500,2601,2704, 2744,2809,2916,3025,3125,3136,3249,3364,3375,3481,3600, 3721,3844,3969,4096,4225,4356,4489,4624,4761,4900,4913, 5041,5184,5329,5476,5625,5776,5832,5929,6084,6241,6400, 6561,6724,6859,6889,7056,7225,7396,7569,7744,7776,7921, 8000,8100,8192,8281,8464,8649,8836,9025,9216,9261,9409, 9604,9801,10000,10201,10404,10609,10648,10816,11025,11236, 11449,11664,11881,12100,12167,12321,12544,12769,12996,13225, 13456,13689,13824,13924,14161,14400,14641,14884,15129,15376, 15625,15876,16129,16384,16641,16807,16900,17161,17424,17576, 17689,17956,18225,18496,18769,19044,19321,19600,19683,19881, 20164,20449,20736,21025,21316,21609,21904,21952,22201,22500, 22801,23104,23409,23716,24025,24336,24389,24649,24964,25281, 25600,25921,26244,26569,26896,27000,27225,27556,27889,28224, 28561,28900,29241,29584,29791,29929,30276,30625,30976,31329, 31684,32041,32400,32761,32768,33124,33489,33856,34225,34596, 34969,35344,35721,35937,36100,36481,36864,37249,37636,38025, 38416,38809,39204,39304,39601,40000,40401,40804,41209,41616, 42025,42436,42849,42875,43264,43681,44100,44521,44944,45369, 45796,46225,46656,47089,47524,47961,48400,48841,49284,49729, 50176,50625,50653,51076,51529,51984,52441,52900,53361,53824, 54289,54756,54872,55225,55696,56169,56644,57121,57600,58081, 58564,59049,59319,59536,60025,60516,61009,61504,62001,62500, 63001,63504,64000,64009,64516,65025,65536,66049,66564,67081, 67600,68121,68644,68921,69169,69696,70225,70756,71289,71824, 72361,72900,73441,73984,74088,74529,75076,75625,76176,76729, 77284,77841,78125,78400,78961,79507,79524,80089,80656,81225, 81796,82369,82944,83521,84100,84681,85184,85264,85849,86436, 87025,87616,88209,88804,89401,90000,90601,91125,91204,91809, 92416,93025,93636,94249,94864,95481,96100,96721,97336,97344, 97969,98596,99225,99856,100000,100489,101124,101761,102400, 103041,103684,103823,104329,104976,105625,106276,106929, 107584,108241,108900,109561,110224,110592,110889,111556, 112225,112896,113569,114244,114921,115600,116281,116964, 117649,118336,119025,119716,120409,121104,121801,122500, 123201,123904,124609,125000,125316,126025,126736,127449, 128164,128881,129600,130321,131044,131072,131769,132496, 132651,133225,133956,134689,135424,136161,136900,137641, 138384,139129,139876,140608,140625,141376,142129,142884, 143641,144400,145161,145924,146689,147456,148225,148877, 148996,149769,150544,151321,152100,152881,153664,154449, 155236,156025,156816,157464,157609,158404,159201,160000, 160801,161051,161604,162409,163216,164025,164836,165649, 166375,166464,167281,168100,168921,169744,170569,171396, 172225,173056,173889,174724,175561,175616,176400,177147, 177241,178084,178929,179776,180625,181476,182329,183184, 184041,184900,185193,185761,186624,187489,188356,189225, 190096,190969,191844,192721,193600,194481,195112,195364, 196249,197136,198025,198916,199809,200704,201601,202500, 203401,204304,205209,205379,206116,207025,207936,208849, 209764,210681,211600,212521,213444,214369,215296,216000, 216225,217156,218089,219024,219961,220900,221841,222784, 223729,224676,225625,226576,226981,227529,228484,229441, 230400,231361,232324,233289,234256,235225,236196,237169, 238144,238328,239121,240100,241081,242064,243049,244036, 245025,246016,247009,248004,248832,249001,250000,250047, 251001,252004,253009,254016,255025,256036,257049,258064, 259081,260100,261121,262144,263169,264196,265225,266256, 267289,268324,269361,270400,271441,272484,273529,274576, 274625,275625,276676,277729,278784,279841,279936,280900, 281961,283024,284089,285156,286225,287296,287496,288369, 289444,290521,291600,292681,293764,294849,295936,297025, 298116,299209,300304,300763,301401,302500,303601,304704, 305809,306916,308025,309136,310249,311364,312481,313600, 314432,314721,315844,316969,318096,319225,320356,321489, 322624,323761,324900,326041,327184,328329,328509,329476, 330625,331776,332929,334084,335241,336400,337561,338724, 339889,341056,342225,343000,343396,344569,345744,346921, 348100,349281,350464,351649,352836,354025,355216,356409, 357604,357911,358801,360000,361201,362404,363609,364816, 366025,367236,368449,369664,370881,371293,372100,373248, 373321,374544,375769,376996,378225,379456,380689,381924, 383161,384400,385641,386884,388129,389017,389376,390625, 391876,393129,394384,395641,396900,398161,399424,400689, 401956,403225,404496,405224,405769,407044,408321,409600, 410881,412164,413449,414736,416025,417316,418609,419904, 421201,421875,422500,423801,425104,426409,427716,429025, 430336,431649,432964,434281,435600,436921,438244,438976, 439569,440896,442225,443556,444889,446224,447561,448900, 450241,451584,452929,454276,455625,456533,456976,458329, 459684,461041,462400,463761,465124,466489,467856,469225, 470596,471969,473344,474552,474721,476100,477481,478864, 480249,481636,483025,484416,485809,487204,488601,490000, 491401,492804,493039,494209,495616,497025,498436,499849, 501264,502681,504100,505521,506944,508369,509796,511225, 512000,512656,514089,515524,516961,518400,519841,521284, 522729,524176,524288,525625,527076,528529,529984,531441, 532900,534361,535824,537289,537824,538756,540225,541696, 543169,544644,546121,547600,549081,550564,551368,552049, 553536,555025,556516,558009,559504,561001,562500,564001, 565504,567009,568516,570025,571536,571787,573049,574564, 576081,577600,579121,580644,582169,583696,585225,586756, 588289,589824,591361,592704,592900,594441,595984,597529, 599076,600625,602176,603729,605284,606841,608400,609961, 611524,613089,614125,614656,616225,617796,619369,620944, 622521,624100,625681,627264,628849,630436,632025,633616, 635209,636056,636804,638401,640000,641601,643204,644809, 646416,648025,649636,651249,652864,654481,656100,657721, 658503,659344,660969,662596,664225,665856,667489,669124, 670761,672400,674041,675684,677329,678976,680625,681472, 682276,683929,685584,687241,688900,690561,692224,693889, 695556,697225,698896,700569,702244,703921,704969,705600, 707281,708964,710649,712336,714025,715716,717409,719104, 720801,722500,724201,725904,727609,729000,729316,731025, 732736,734449,736164,737881,739600,741321,743044,744769, 746496,748225,749956,751689,753424,753571,755161,756900, 758641,759375,760384,762129,763876,765625,767376,769129, 770884,772641,774400,776161,777924,778688,779689,781456, 783225,784996,786769,788544,790321,792100,793881,795664, 797449,799236,801025,802816,804357,804609,806404,808201, 810000,811801,813604,815409,817216,819025,820836,822649, 823543,824464,826281,828100,829921,830584,831744,833569, 835396,837225,839056,840889,842724,844561,846400,848241, 850084,851929,853776,855625,857375,857476,859329,861184, 863041,864900,866761,868624,870489,872356,874225,876096, 877969,879844,881721,883600,884736,885481,887364,889249, 891136,893025,894916,896809,898704,900601,902500,904401, 906304,908209,910116,912025,912673,913936,915849,917764, 919681,921600,923521,925444,927369,929296,931225,933156, 935089,937024,938961,940900,941192,942841,944784,946729, 948676,950625,952576,954529,956484,958441,960400,962361, 964324,966289,968256,970225,970299,972196,974169,976144, 978121,980100,982081,984064,986049,988036,990025,992016, 994009,996004,998001,1000000 I'll use this data to make some plots. Last edited by krassi_holmz (20060521 19:10:48) IPBLE: Increasing Performance By Lowering Expectations. #4 20060521 19:12:48
Re: Cool Perfect Power Function!!!Very interesting. Last edited by krassi_holmz (20060521 19:13:39) IPBLE: Increasing Performance By Lowering Expectations. #5 20060521 19:14:37
Re: Cool Perfect Power Function!!!Here's a plot up to 1000: Last edited by krassi_holmz (20060521 19:15:21) IPBLE: Increasing Performance By Lowering Expectations. #6 20060521 19:15:55
Re: Cool Perfect Power Function!!!It looks familiar. IPBLE: Increasing Performance By Lowering Expectations. #7 20060521 19:28:45
Re: Cool Perfect Power Function!!!Look: IPBLE: Increasing Performance By Lowering Expectations. #8 20060521 19:35:09
Re: Cool Perfect Power Function!!!Now some pure maths: 2. find the nmber of all perfect powers less or equal to x. 3. What can you say about the increasing of P? I think it diverges.(the increasing, not the function, although F diverges too). Last edited by krassi_holmz (20060521 19:36:17) IPBLE: Increasing Performance By Lowering Expectations. #9 20060521 19:52:05
Re: Cool Perfect Power Function!!!Let means , where p_i is the n=th prime. IPBLE: Increasing Performance By Lowering Expectations. #10 20060521 20:05:03
Re: Cool Perfect Power Function!!!Now, if (1). p divides i with probability 1/p, so the probability that (1) is tue, is 1/p^n. So the probability x to be a perfect power is: , where P is the set of prime numbers. But now, every i may be every number between 0 and infty and the resulting x will be always different, so if is the set of all numbers that are divisible only by , then exactly of them will be perfect powers. Now, when n goes to infty, we get that the distribution of the perfect powers is in the set of the integers is: . Now that's a hard sum!!! (it will be < 1, because ) Last edited by krassi_holmz (20060521 20:16:25) IPBLE: Increasing Performance By Lowering Expectations. #11 20060521 20:23:16
Re: Cool Perfect Power Function!!!Now, how to find ??? Any ideas? (and you can disscuss my work and report for errors) IPBLE: Increasing Performance By Lowering Expectations. #12 20060522 01:16:56
Re: Cool Perfect Power Function!!!I see I'm not the only one who has conversations with himself. A logarithm is just a misspelled algorithm. #13 20060522 01:52:55
Re: Cool Perfect Power Function!!!
i only usually do when im figuring something out and asking at the same time, and then end up answering myself before anyone responds The Beginning Of All Things To End. The End Of All Things To Come. #14 20060522 01:59:51
Re: Cool Perfect Power Function!!!
No function defined on the integers (or naturals as is this case) can be continuous or differentiable. For any function to be continuous at any given point, that point must be a limit point. In functions of integers, there are no limit points. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #15 20060522 03:32:50
Re: Cool Perfect Power Function!!!
Right, I know that, I just wasn't sure if krassi had developed a formula for the intermediate points. A logarithm is just a misspelled algorithm. #16 20060522 04:20:26
Re: Cool Perfect Power Function!!!To tell the truth i think we all talk to our selves.. (after 3 or 4 Dr. Peppers i know i do...) Do you like my new avatar? (even if you don't i do ) (it's victor from oh my gods) An Incremental Development #17 20060523 10:12:46
Re: Cool Perfect Power Function!!!kras, why no 'L' in this line?: const long int n=1000000L; igloo myrtilles fourmis #18 20060523 11:26:15
Re: Cool Perfect Power Function!!!In most compilers, long is the same as int, 32 bits. So it wouldn't matter. Long is (right now) an artifact of when you had 16 bit ints and 32 bit longs. But now ints are 32 bits, and having a 64 bit int would make a heck of a lot of more work for the processor, it's really not needed. Although compilers normally also have 64 bit ints like microsoft's INT_64. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #19 20060523 21:15:14
Re: Cool Perfect Power Function!!!Since p is a prime number, how can it divide i? Last edited by George,Y (20060523 21:15:48) X'(yXβ)=0 #20 20060523 22:51:48
Re: Cool Perfect Power Function!!!
ok. 32bit integers. IPBLE: Increasing Performance By Lowering Expectations. #21 20060523 23:01:05
Re: Cool Perfect Power Function!!!3 is a prime. IPBLE: Increasing Performance By Lowering Expectations. #22 20060524 00:39:53
Re: Cool Perfect Power Function!!!kras, the biggest number I got to work for n is: igloo myrtilles fourmis #23 20060524 00:49:25
Re: Cool Perfect Power Function!!!Yes. But why? IPBLE: Increasing Performance By Lowering Expectations. #24 20060524 01:20:41
Re: Cool Perfect Power Function!!!kras, I recompiled with a different C compiler, and now it Kraps out on this number: Last edited by John E. Franklin (20060524 01:21:31) igloo myrtilles fourmis #25 20060524 01:22:33
Re: Cool Perfect Power Function!!!We have to do something about this foul word changer. igloo myrtilles fourmis 