20120502, 21:51  #1 
(loop (#_fork))
Feb 2006
Cambridge, England
3^{3}×239 Posts 
Sixbrilliant numbers
That is, numbers with six prime factors all of the same number of digits. Ones marked with dots are bydefinition otherwise by search; there are obviously no 7digit without dupes.
Code:
.1771561 11,11,11,11,11,11 smallest 7digit with dupes 9887999 11,11,11,17,19,23 largest 7digit with dupes 10081357 11,11,13,13,17,29 smallest 8digit with dupes .30808063 11,13,17,19,23,29 smallest 8digit without dupes 99423181 11,13,19,23,37,43 largest 8digit without dupes 99990407 11,11,19,23,31,61 largest 8digit with dupes 100016059 11,11,13,13,67,73 smallest 9digit with dupes 100145903 11,13,19,29,31,41 smallest 9digit without dupes 999978551 11,13,29,51,61,67 largest 9digit with or without dupes 1000052443 13,23,29,29,41,97 smallest 10digit with dupes 1000202489 11,13,29,43,71,79 smallest 10digit without dupes 9999884737 19,31,43,67,71,83 largest 10digit with or without dupes 10000108889 23,29,59,59,59,73 smallest 11digit with dupes 10000268641 11,31,43,79,89,97 smallest 11digit without dupes 99919838357 53,61,67,71,73,89 largest 11digit without dupes 99998113069 47,61,61,83,83,83 largest 11digit with dupes 100000197487 41,53,67,73,97,97 smallest 12digit with dupes 100042414319 53,59,61,71,83,89 smallest 12digit without dupes .293391909323 71,73,79,83,89,97 largest 12digit without dupes .832972004929 97,97,97,97,97,97 largest 12digit with dupes Last fiddled with by fivemack on 20120502 at 21:56 
20120502, 22:51  #2 
(loop (#_fork))
Feb 2006
Cambridge, England
14465_{8} Posts 
Code:
1061520150601 101 101 101 101 101 101 with (!) 1741209542339 101 103 107 109 113 127 without (!) 9999901067561 101 107 109 139 157 389 without 9999965520509 101 107 113 197 197 211 with 10000001342381 101 101 149 163 181 223 with 10000087934327 101 109 131 149 173 269 without 99999998106163 103 149 179 223 239 683 without 99999999253139 113 181 199 199 331 373 with 100000001274419 101 101 151 211 313 983 with 100000002228881 107 109 113 307 439 563 without 999999998566003 131 167 191 379 653 967 with/without 1000000000407943 151 311 317 379 421 421 with 1000000002307693 109 163 191 593 653 761 without 9999999998178521 131 353 467 647 751 953 with/without 10000000002328283 151 397 467 599 691 863 with/without 99999999886001279 359 631 673 821 877 911 with/without 100000000103742353 563 577 673 751 751 811 with 100000000374707123 401 449 761 773 947 997 without 890969009638765049 967 971 977 983 991 997 without(!) 982134461213542729 997 997 997 997 997 997 with (!) Code:
#include <vector> #include <algorithm> #include <cmath> #include <cstdio> #include <iostream> using std::vector; using std::sort; using std::cout; using std::endl; bool isprime(int T) { if (T==2  T==3) return true; if (T%2==0) return false; if (T%3==0) return false; int s=5,ds=2; while (s*s<=T) { if (T%s==0) return false; s+=ds; ds=6ds; } return true; } int main(void) { vector<double> logthreep; vector<int> primes; for (int i=100; i<1000; i++) if(isprime(i)) primes.push_back(i); int pn = primes.size(); for (int pi1=0; pi1<pn; pi1++) for (int pi2=pi1; pi2<pn; pi2++) for (int pi3=pi2; pi3<pn; pi3++) { logthreep.push_back(log(primes[pi1])+log(primes[pi2])+log(primes[pi3])); } cout << logthreep.size() << "\n\n"; sort(logthreep.begin(), logthreep.end()); double target = 14.0*log(10); double XXH,XXH1,XXH2,XXL,XXL1,XXL2; XXH=XXH1=XXH2=1e9; XXL=XXL1=XXL2=1e9; for (int q=0; q<logthreep.size(); q++) { double t2 = targetlogthreep[q]; vector<double>::iterator ilo,ihi; ihi = lower_bound(logthreep.begin(), logthreep.end(), t2); if (ihi != logthreep.begin()) { ilo = ihi1; if (*ilo!=0) { if (logthreep[q]+*ilo > XXL) { XXL1=logthreep[q]; XXL2=*ilo; XXL=XXL1+XXL2; } if (logthreep[q]+*ihi < XXH) { XXH1=logthreep[q]; XXH2=*ihi; XXH=XXH1+XXH2; } long long U0 = (int)(0.1+exp(logthreep[q])); long long U1 = (int)(0.1+exp(*ilo)); long long U2 = (int)(0.1+exp(*ihi)); cout << q << " " << U0 << " " << U1 << " " << U2 << " " << U0*U1 << " " << U0*U2 << endl; cout << q << " " << (int)(0.1+exp(*ilo)) << " " << (int)(0.1+exp(*ihi)) << endl; } } } printf("%30.28f %30.28f\n%30.28f %30.28f\n",exp(XXL1),exp(XXL2),exp(XXH1),exp(XXH2)); } Code:
./meeble awk 'NF==6 && $4!=1'  sort gk6awk '{print $6}'  uniq c  head Last fiddled with by fivemack on 20120502 at 22:52 
20120502, 23:05  #3 
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
I'm guessing:
Code:
isbrilliant(x,y)=for(z=2^x,10^y,if(sum(a=1,if(#factor(z)[,2]<6,#factor(z)[,2],break()),factor(z)[a,2])==6,print(z))) 
20120502, 23:39  #4  
(loop (#_fork))
Feb 2006
Cambridge, England
3^{3}×239 Posts 
Quote:
The C++ sortandsearch code I posted gives the wrong answer for the 22digit case, because 53bit doubles aren't wide enough, though obviously the answer from the C++ code bounds the size of the area you'd have to sieve. Last fiddled with by fivemack on 20120502 at 23:40 

20120503, 00:08  #5  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:


20120503, 00:35  #6  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Quote:
(Excuse the poor tex, it may or may not be better than "GCD(1000#/100#, n)". ) Last fiddled with by Dubslow on 20120503 at 00:43 Reason: tex tex tex tex tex 

20120503, 01:15  #7  
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
Quote:
okay never mind it's >202 that it tells you anything because it's the first multiple of a prime with 3 or more digits. Last fiddled with by science_man_88 on 20120503 at 01:27 

20120503, 01:22  #8  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
Quote:
"numbers with six prime factors all of the same number of digits" 729 isn't prime; 3 is certainly not a 3 digit prime. And last time I checked, no 3 digit number will have 6 3digit prime factors. Also last time I checked, if you do GCD(101, a number with 101 in its factorization), you'll get at least 101. I fail to see what 300 has to do with that. 

20120503, 01:33  #9  
"Forget I exist"
Jul 2009
Dumbassville
10000011000000_{2} Posts 
Quote:
Code:
(22:31)>isbrilliant(6,3,4) 144 160 216 224 240 324 336 352 360 400 416 486 504 528 540 544 560 600 608 624 729 736 756 784 792 810 816 840 880 900 912 928 936 992 1000 Code:
for(n=100,999,print(gcd(prod(X=26,68,prime(X))^6,n)","n)) Last fiddled with by science_man_88 on 20120503 at 01:55 

20120503, 02:25  #10 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
7221_{10} Posts 

20120503, 06:56  #11  
Aug 2006
1011101011011_{2} Posts 
Quote:
The counts of such numbers are not in the OEIS, but the counts for the analogous semiprimes are A087434. The formula is straightforward and gives Code:
84, 230230, 13172431272, 2009505083440218, 476013062485109252824, 148697710093074782891561317, 56289121765869792295257792982777, 24349676137284030200879331533337194376, 11666012947107695154385353794415395985137344, 6057277355183085204144847469602671151899349499197, 3354985557158369214407984988027190131369315871722691152, 1959509071991741722270413421131623309713272464704021335869310, 1196301271416935612331559019142578123179924493160149725594041285390, 758297477798207294085616092160894284729546088426880198004979229145638184, 496405409134346365011354234191682214886538111380694256570445787621168130062528, 334180505840507943479437772268647724355589483819448761562142733090916595143509225212, 230551283386125977830557515195210002315457614489571967352960948012129577632233414552542796, 162537577146540321845227986334505185151917953317422163374965016524615696383252603524891046763532, 116816284673708680421835123424581599081284210871219523674771358297205544299884690188130918699786791500, 85416640764723824021108446456250433057219850414437417656346074325343176420847706796002701169332531133021551, 63434778901223585384265131927059356802305353342435316304189778022372093879998625701188622017546319272210868240784, 47777139494090360016305651100722437896951187322404498059597560979956679547504643361209098188623110927359491777939720841, 36447608389806648140714977201273896196346361099368631002618281535486043280009949162450062261360663543889016612413795952580291, 28131574280009154082710448682314774375712261908758572574980667541365874655616083694412009836355371649813984129660663498677543353152, ... 

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