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June 12

Unique close factors

For a number n, call a and b close factors of n if a*b =n and a<b<2*a. define C(k) as the lowest number with k different pairs of close factors. So C(1) = 2*3 = 6. I think C(2) = 432 = 27*16 = 24*18. Any idea if C(2) is correct (or if there is a lower number) and any ideas how to calculate C(k) for k>2. — Preceding unsigned comment added by Naraht (talk • contribs)

120 = 8*15 = 10*12. We have

a^{2}<n<2a^{2}

so

{\sqrt {\frac {n}{2}}}<a<{\sqrt {n}}

so a brute force method of determining how many close divisor-pairs n has is to check each integer between sqrt(n/2) and sqrt(n) and count how many are divisors of n. For large values of n this becomes increasingly inefficient, and factorising n to determine a complete list of its divisors becomes more attractive. Gandalf61 (talk) 08:27, 12 June 2012 (UTC)[reply]

Thank you. I had missed 120. I was looking for pairs that "traded" a 2 for a 3, your example "trades" a 4 for a 5. However, it may be very easy to limit the possible numbers to check, since a "trade" between two pairs would involve multiplying by one number and dividing by another each term. This probably puts a minimum based on k of the number of different factors that C(k) would have. C(3) would have to say have one pair trade a 4 for a 5, one trade a 5 for a 6 and one trade a 4 for a 6 or something similar. I think C(3) *trades* 5, 6 & 7 around with 1680 = 30*56 = 35*48 = 40*42. C(4) would have to have 6 different trades. Hmm...Naraht (talk) 17:52, 12 June 2012 (UTC).[reply]

Also, both 120 and 1680 being one less than a square are of the form n * (n+2) with both n and n+2 having a relatively high number of factors. This makes sense since having one pair of close factors almost as close as possible would be good.Naraht (talk) 17:55, 12 June 2012 (UTC)[reply]

Here is a naive brute force PARI/GP search for terms below 10⁸:

found=vector(100);
{for(n=1,10^8,
  c=0;
  fordiv(n,a,b=n/a;if(a<b&b<2*a,c++));
  if(c>0&!found[c],
    print1(":C(",c,") = ",n);
    fordiv(n,a,b=n/a;if(a<b&b<2*a,print1(" = ",a,"*",b)));
    print();
    found[c]=1;
  )
)}

It took an hour at 2.4 GHz to compute:

Results to 10⁸

C(1) = 6 = 2*3

C(2) = 120 = 8*15 = 10*12

C(3) = 840 = 21*40 = 24*35 = 28*30

C(4) = 2520 = 36*70 = 40*63 = 42*60 = 45*56

C(5) = 10080 = 72*140 = 80*126 = 84*120 = 90*112 = 96*105

C(6) = 27720 = 120*231 = 126*220 = 132*210 = 140*198 = 154*180 = 165*168

C(7) = 50400 = 160*315 = 168*300 = 175*288 = 180*280 = 200*252 = 210*240 = 224*225

C(8) = 83160 = 210*396 = 216*385 = 220*378 = 231*360 = 252*330 = 264*315 = 270*308 = 280*297

C(9) = 138600 = 264*525 = 275*504 = 280*495 = 300*462 = 308*450 = 315*440 = 330*420 = 350*396 = 360*385

C(10) = 277200 = 385*720 = 396*700 = 400*693 = 420*660 = 440*630 = 450*616 = 462*600 = 495*560 = 504*550 = 525*528

C(11) = 360360 = 429*840 = 440*819 = 455*792 = 462*780 = 468*770 = 495*728 = 504*715 = 520*693 = 546*660 = 572*630 = 585*616

C(13) = 720720 = 616*1170 = 624*1155 = 630*1144 = 660*1092 = 693*1040 = 715*1008 = 720*1001 = 728*990 = 770*936 = 780*924 = 792*910 = 819*880 = 840*858

C(12) = 831600 = 660*1260 = 675*1232 = 693*1200 = 700*1188 = 720*1155 = 756*1100 = 770*1080 = 792*1050 = 825*1008 = 840*990 = 880*945 = 900*924

C(15) = 1441440 = 858*1680 = 880*1638 = 910*1584 = 924*1560 = 936*1540 = 990*1456 = 1001*1440 = 1008*1430 = 1040*1386 = 1056*1365 = 1092*1320 = 1120*1287 = 1144*1260 = 1155*1248 = 1170*1232

C(14) = 1663200 = 924*1800 = 945*1760 = 990*1680 = 1008*1650 = 1050*1584 = 1056*1575 = 1080*1540 = 1100*1512 = 1120*1485 = 1155*1440 = 1188*1400 = 1200*1386 = 1232*1350 = 1260*1320

C(16) = 2162160 = 1040*2079 = 1080*2002 = 1092*1980 = 1144*1890 = 1155*1872 = 1170*1848 = 1188*1820 = 1232*1755 = 1260*1716 = 1287*1680 = 1320*1638 = 1365*1584 = 1386*1560 = 1404*1540 = 1430*1512 = 1456*1485

C(17) = 3326400 = 1320*2520 = 1344*2475 = 1350*2464 = 1386*2400 = 1400*2376 = 1440*2310 = 1485*2240 = 1512*2200 = 1540*2160 = 1575*2112 = 1584*2100 = 1600*2079 = 1650*2016 = 1680*1980 = 1728*1925 = 1760*1890 = 1800*1848

C(18) = 4324320 = 1485*2912 = 1512*2860 = 1540*2808 = 1560*2772 = 1584*2730 = 1638*2640 = 1680*2574 = 1716*2520 = 1755*2464 = 1760*2457 = 1820*2376 = 1848*2340 = 1872*2310 = 1890*2288 = 1980*2184 = 2002*2160 = 2016*2145 = 2079*2080

C(21) = 7207200 = 1925*3744 = 1950*3696 = 1980*3640 = 2002*3600 = 2016*3575 = 2080*3465 = 2100*3432 = 2145*3360 = 2184*3300 = 2200*3276 = 2275*3168 = 2288*3150 = 2310*3120 = 2340*3080 = 2400*3003 = 2464*2925 = 2475*2912 = 2520*2860 = 2574*2800 = 2600*2772 = 2640*2730

C(22) = 8648640 = 2080*4158 = 2112*4095 = 2145*4032 = 2160*4004 = 2184*3960 = 2240*3861 = 2288*3780 = 2310*3744 = 2340*3696 = 2376*3640 = 2457*3520 = 2464*3510 = 2496*3465 = 2520*3432 = 2574*3360 = 2640*3276 = 2730*3168 = 2772*3120 = 2808*3080 = 2860*3024 = 2880*3003 = 2912*2970

C(20) = 9424800 = 2200*4284 = 2244*4200 = 2310*4080 = 2380*3960 = 2400*3927 = 2448*3850 = 2464*3825 = 2475*3808 = 2520*3740 = 2550*3696 = 2618*3600 = 2640*3570 = 2720*3465 = 2772*3400 = 2800*3366 = 2805*3360 = 2856*3300 = 2975*3168 = 2992*3150 = 3060*3080

C(19) = 9979200 = 2240*4455 = 2268*4400 = 2310*4320 = 2376*4200 = 2400*4158 = 2464*4050 = 2475*4032 = 2520*3960 = 2592*3850 = 2640*3780 = 2700*3696 = 2772*3600 = 2800*3564 = 2835*3520 = 2880*3465 = 2970*3360 = 3024*3300 = 3080*3240 = 3150*3168

C(23) = 14414400 = 2730*5280 = 2772*5200 = 2800*5148 = 2860*5040 = 2880*5005 = 2912*4950 = 2925*4928 = 3003*4800 = 3080*4680 = 3120*4620 = 3150*4576 = 3168*4550 = 3276*4400 = 3300*4368 = 3360*4290 = 3432*4200 = 3465*4160 = 3520*4095 = 3575*4032 = 3600*4004 = 3640*3960 = 3696*3900 = 3744*3850

C(26) = 21621600 = 3300*6552 = 3360*6435 = 3432*6300 = 3465*6240 = 3510*6160 = 3575*6048 = 3600*6006 = 3640*5940 = 3696*5850 = 3744*5775 = 3780*5720 = 3850*5616 = 3861*5600 = 3900*5544 = 3960*5460 = 4004*5400 = 4095*5280 = 4158*5200 = 4200*5148 = 4290*5040 = 4320*5005 = 4368*4950 = 4400*4914 = 4550*4752 = 4576*4725 = 4620*4680

C(25) = 24504480 = 3536*6930 = 3570*6864 = 3640*6732 = 3696*6630 = 3740*6552 = 3744*6545 = 3808*6435 = 3927*6240 = 3960*6188 = 3978*6160 = 4004*6120 = 4080*6006 = 4095*5984 = 4284*5720 = 4290*5712 = 4368*5610 = 4420*5544 = 4488*5460 = 4576*5355 = 4620*5304 = 4641*5280 = 4680*5236 = 4760*5148 = 4862*5040 = 4896*5005

C(27) = 32432400 = 4050*8008 = 4095*7920 = 4158*7800 = 4200*7722 = 4212*7700 = 4290*7560 = 4368*7425 = 4400*7371 = 4455*7280 = 4536*7150 = 4550*7128 = 4620*7020 = 4680*6930 = 4725*6864 = 4752*6825 = 4914*6600 = 4950*6552 = 5005*6480 = 5040*6435 = 5148*6300 = 5200*6237 = 5265*6160 = 5400*6006 = 5460*5940 = 5544*5850 = 5616*5775 = 5670*5720

C(24) = 33153120 = 4095*8096 = 4140*8008 = 4186*7920 = 4290*7728 = 4368*7590 = 4485*7392 = 4554*7280 = 4576*7245 = 4620*7176 = 4680*7084 = 4784*6930 = 4830*6864 = 5005*6624 = 5040*6578 = 5060*6552 = 5148*6440 = 5152*6435 = 5280*6279 = 5313*6240 = 5382*6160 = 5460*6072 = 5520*6006 = 5544*5980 = 5720*5796

C(28) = 36756720 = 4290*8568 = 4368*8415 = 4420*8316 = 4488*8190 = 4590*8008 = 4620*7956 = 4641*7920 = 4680*7854 = 4752*7735 = 4760*7722 = 4862*7560 = 4914*7480 = 5005*7344 = 5040*7293 = 5049*7280 = 5148*7140 = 5236*7020 = 5304*6930 = 5355*6864 = 5460*6732 = 5544*6630 = 5610*6552 = 5616*6545 = 5712*6435 = 5720*6426 = 5940*6188 = 5967*6160 = 6006*6120

C(30) = 43243200 = 4680*9240 = 4725*9152 = 4752*9100 = 4800*9009 = 4914*8800 = 4928*8775 = 4950*8736 = 5005*8640 = 5040*8580 = 5148*8400 = 5200*8316 = 5280*8190 = 5400*8008 = 5460*7920 = 5544*7800 = 5600*7722 = 5616*7700 = 5720*7560 = 5775*7488 = 5824*7425 = 5850*7392 = 5940*7280 = 6006*7200 = 6048*7150 = 6160*7020 = 6240*6930 = 6300*6864 = 6336*6825 = 6435*6720 = 6552*6600

C(29) = 49008960 = 5005*9792 = 5040*9724 = 5148*9520 = 5236*9360 = 5280*9282 = 5304*9240 = 5355*9152 = 5440*9009 = 5460*8976 = 5544*8840 = 5610*8736 = 5712*8580 = 5720*8568 = 5824*8415 = 5984*8190 = 6006*8160 = 6120*8008 = 6160*7956 = 6188*7920 = 6240*7854 = 6336*7735 = 6435*7616 = 6545*7488 = 6552*7480 = 6630*7392 = 6720*7293 = 6732*7280 = 6864*7140 = 6930*7072

C(31) = 61261200 = 5544*11050 = 5610*10920 = 5712*10725 = 5720*10710 = 5775*10608 = 5850*10472 = 5950*10296 = 6006*10200 = 6120*10010 = 6160*9945 = 6188*9900 = 6300*9724 = 6435*9520 = 6545*9360 = 6552*9350 = 6600*9282 = 6630*9240 = 6732*9100 = 6800*9009 = 6825*8976 = 6864*8925 = 6930*8840 = 7140*8580 = 7150*8568 = 7280*8415 = 7293*8400 = 7480*8190 = 7650*8008 = 7700*7956 = 7735*7920 = 7800*7854

C(32) = 73513440 = 6120*12012 = 6160*11934 = 6188*11880 = 6240*11781 = 6426*11440 = 6435*11424 = 6545*11232 = 6552*11220 = 6630*11088 = 6732*10920 = 6864*10710 = 6930*10608 = 7020*10472 = 7072*10395 = 7140*10296 = 7280*10098 = 7293*10080 = 7344*10010 = 7392*9945 = 7480*9828 = 7560*9724 = 7722*9520 = 7735*9504 = 7854*9360 = 7920*9282 = 7956*9240 = 8008*9180 = 8160*9009 = 8190*8976 = 8316*8840 = 8415*8736 = 8568*8580

C(33) = 86486400 = 6600*13104 = 6720*12870 = 6825*12672 = 6864*12600 = 6930*12480 = 7020*12320 = 7040*12285 = 7150*12096 = 7200*12012 = 7280*11880 = 7392*11700 = 7425*11648 = 7488*11550 = 7560*11440 = 7700*11232 = 7722*11200 = 7800*11088 = 7920*10920 = 8008*10800 = 8064*10725 = 8190*10560 = 8316*10400 = 8320*10395 = 8400*10296 = 8580*10080 = 8640*10010 = 8736*9900 = 8775*9856 = 8800*9828 = 9009*9600 = 9100*9504 = 9152*9450 = 9240*9360

It assumes C(k) must have exactly k pairs. If it must have at least k then skip the terms which are out of order so for example C(12) = C(13) = 720720. PrimeHunter (talk) 11:08, 15 June 2012 (UTC)[reply]

Wow! and Thank You! Definitely the right tool. And I agree that it is Naive, I think anything beyond C(5) or C(6) would *have* to be divisible by 30, but proving it might be a little tricky. I'n sort of surprised at how different the 'exactly k pairs' and 'at least k pairs' are, the fact that for 'at least k pairs' C(25), C(26), C(27) and C(28) are the same. All of the numbers are *highly* abundant, but it doesn't appear that the list matches any highly abundant (or related) lists on wikipedia, in fact, neither the exactly or at least sequences exists in OEIS (nothing has 6,12,840,2520). Hmm. I wonder if I can get away with installing PARI/GP on our development box at work.Naraht (talk) 15:14, 15 June 2012 (UTC)[reply]

The tool works and is easy to use but it isn't fast. Below is a better algorithm in the faster C (programming language).

#include <stdio.h>
#include <math.h>
#define maxn 100000000
unsigned char count[maxn+1], found[100];
unsigned int a,c,n,ab,maxab,maxa;

int main(int argc, char *argv[]) {
  maxa=sqrt(maxn);
  for(a=1; a<=maxa; a++) {
    maxab=a*(2*a-1);
    if (maxab>maxn) maxab=maxn;
    for(ab=a*(a+1); ab<=maxab; ab+=a) count[ab]++;
  }
  for(n=1; n<=maxn; n++) {
    c=count[n];
    if(c&&!found[c]) {
      printf(":C(%u) = %u",c,n);
      maxa=sqrt(n-0.5);
      for(a=sqrt(n/2-0.5)+1; a<=maxa; a++)
        if (n%a==0) printf(" = %u*%u",a,n/a);
      printf("\n");
      found[c]=1;
    }
  }
  return 0;
}

It makes the same output but only takes 2 seconds on the same cpu. The algorithm scales better. Changing to maxn = 1000000000 (10⁹) only takes 20 seconds. After that it slows down due to memory usage and only works to around 1.8*10⁹ where it runs out of ram on my PC. It adds these results from 10⁸ to 1.8*10⁹:

Results from 10⁸ to 1.8*10⁹

C(36) = 122522400 = 7854*15600 = 7920*15470 = 7956*15400 = 8008*15300 = 8160*15015 = 8190*14960 = 8400*14586 = 8415*14560 = 8568*14300 = 8580*14280 = 8736*14025 = 8800*13923 = 8840*13860 = 8925*13728 = 8976*13650 = 9009*13600 = 9100*13464 = 9240*13260 = 9282*13200 = 9350*13104 = 9360*13090 = 9520*12870 = 9724*12600 = 9900*12376 = 9945*12320 = 10010*12240 = 10080*12155 = 10200*12012 = 10296*11900 = 10400*11781 = 10472*11700 = 10608*11550 = 10710*11440 = 10725*11424 = 10920*11220 = 11050*11088

C(34) = 123243120 = 7920*15561 = 7980*15444 = 8008*15390 = 8151*15120 = 8190*15048 = 8208*15015 = 8316*14820 = 8360*14742 = 8424*14630 = 8580*14364 = 8645*14256 = 8778*14040 = 8892*13860 = 8910*13832 = 9009*13680 = 9072*13585 = 9240*13338 = 9360*13167 = 9405*13104 = 9576*12870 = 9828*12540 = 9880*12474 = 10010*12312 = 10032*12285 = 10260*12012 = 10296*11970 = 10374*11880 = 10395*11856 = 10530*11704 = 10640*11583 = 10773*11440 = 10868*11340 = 10920*11286 = 11088*11115

C(35) = 129729600 = 8100*16016 = 8190*15840 = 8316*15600 = 8400*15444 = 8424*15400 = 8580*15120 = 8640*15015 = 8736*14850 = 8775*14784 = 8800*14742 = 8910*14560 = 9009*14400 = 9072*14300 = 9100*14256 = 9152*14175 = 9240*14040 = 9360*13860 = 9450*13728 = 9504*13650 = 9828*13200 = 9900*13104 = 10010*12960 = 10080*12870 = 10296*12600 = 10395*12480 = 10400*12474 = 10530*12320 = 10560*12285 = 10725*12096 = 10800*12012 = 10920*11880 = 11088*11700 = 11200*11583 = 11232*11550 = 11340*11440

C(38) = 147026880 = 8580*17136 = 8640*17017 = 8736*16830 = 8840*16632 = 8976*16380 = 9009*16320 = 9152*16065 = 9180*16016 = 9240*15912 = 9282*15840 = 9360*15708 = 9504*15470 = 9520*15444 = 9724*15120 = 9792*15015 = 9828*14960 = 9945*14784 = 10010*14688 = 10080*14586 = 10098*14560 = 10296*14280 = 10395*14144 = 10472*14040 = 10560*13923 = 10608*13860 = 10710*13728 = 10920*13464 = 11088*13260 = 11220*13104 = 11232*13090 = 11424*12870 = 11440*12852 = 11781*12480 = 11880*12376 = 11934*12320 = 11968*12285 = 12012*12240 = 12096*12155

C(37) = 151351200 = 8736*17325 = 8775*17248 = 8800*17199 = 8820*17160 = 9009*16800 = 9100*16632 = 9240*16380 = 9360*16170 = 9450*16016 = 9504*15925 = 9555*15840 = 9702*15600 = 9800*15444 = 9828*15400 = 9900*15288 = 10010*15120 = 10080*15015 = 10192*14850 = 10296*14700 = 10395*14560 = 10400*14553 = 10584*14300 = 10725*14112 = 10780*14040 = 10800*14014 = 10920*13860 = 11025*13728 = 11088*13650 = 11232*13475 = 11440*13230 = 11466*13200 = 11550*13104 = 11700*12936 = 11760*12870 = 11880*12740 = 12012*12600 = 12285*12320

C(39) = 183783600 = 9724*18900 = 9828*18700 = 9900*18564 = 9945*18480 = 10010*18360 = 10098*18200 = 10200*18018 = 10296*17850 = 10395*17680 = 10472*17550 = 10608*17325 = 10710*17160 = 10725*17136 = 10800*17017 = 10920*16830 = 11050*16632 = 11088*16575 = 11220*16380 = 11440*16065 = 11475*16016 = 11550*15912 = 11700*15708 = 11781*15600 = 11880*15470 = 11900*15444 = 11934*15400 = 12012*15300 = 12155*15120 = 12240*15015 = 12285*14960 = 12376*14850 = 12600*14586 = 12852*14300 = 12870*14280 = 13090*14040 = 13104*14025 = 13200*13923 = 13260*13860 = 13464*13650

C(40) = 205405200 = 10260*20020 = 10296*19950 = 10374*19800 = 10395*19760 = 10450*19656 = 10640*19305 = 10725*19152 = 10800*19019 = 10868*18900 = 10920*18810 = 11088*18525 = 11115*18480 = 11286*18200 = 11400*18018 = 11440*17955 = 11550*17784 = 11700*17556 = 11704*17550 = 11856*17325 = 11880*17290 = 11970*17160 = 12012*17100 = 12285*16720 = 12350*16632 = 12540*16380 = 12600*16302 = 12825*16016 = 12870*15960 = 13104*15675 = 13167*15600 = 13200*15561 = 13300*15444 = 13338*15400 = 13585*15120 = 13650*15048 = 13680*15015 = 13832*14850 = 13860*14820 = 14040*14630 = 14300*14364

C(42) = 245044800 = 11088*22100 = 11200*21879 = 11220*21840 = 11424*21450 = 11440*21420 = 11550*21216 = 11700*20944 = 11781*20800 = 11900*20592 = 11968*20475 = 12012*20400 = 12155*20160 = 12240*20020 = 12320*19890 = 12376*19800 = 12480*19635 = 12600*19448 = 12870*19040 = 13090*18720 = 13104*18700 = 13200*18564 = 13260*18480 = 13464*18200 = 13600*18018 = 13650*17952 = 13728*17850 = 13860*17680 = 13923*17600 = 14025*17472 = 14144*17325 = 14280*17160 = 14300*17136 = 14400*17017 = 14560*16830 = 14586*16800 = 14784*16575 = 14960*16380 = 15015*16320 = 15300*16016 = 15400*15912 = 15470*15840 = 15600*15708

C(41) = 273873600 = 11704*23400 = 11856*23100 = 11970*22880 = 12012*22800 = 12320*22230 = 12350*22176 = 12480*21945 = 12540*21840 = 12600*21736 = 12768*21450 = 12870*21280 = 13104*20900 = 13167*20800 = 13200*20748 = 13300*20592 = 13376*20475 = 13585*20160 = 13650*20064 = 13680*20020 = 13728*19950 = 13832*19800 = 13860*19760 = 14300*19152 = 14400*19019 = 14560*18810 = 14630*18720 = 14784*18525 = 14820*18480 = 15015*18240 = 15048*18200 = 15200*18018 = 15400*17784 = 15561*17600 = 15600*17556 = 15675*17472 = 15808*17325 = 15840*17290 = 15960*17160 = 16016*17100 = 16302*16800 = 16380*16720

C(48) = 367567200 = 13600*27027 = 13650*26928 = 13728*26775 = 13860*26520 = 13923*26400 = 14025*26208 = 14040*26180 = 14280*25740 = 14300*25704 = 14560*25245 = 14586*25200 = 14688*25025 = 14850*24752 = 14960*24570 = 15015*24480 = 15120*24310 = 15300*24024 = 15400*23868 = 15444*23800 = 15470*23760 = 15600*23562 = 15708*23400 = 15840*23205 = 15912*23100 = 16016*22950 = 16065*22880 = 16380*22440 = 16575*22176 = 16632*22100 = 16800*21879 = 16830*21840 = 17017*21600 = 17136*21450 = 17160*21420 = 17325*21216 = 17550*20944 = 17680*20790 = 17850*20592 = 17952*20475 = 18018*20400 = 18200*20196 = 18360*20020 = 18480*19890 = 18564*19800 = 18700*19656 = 18720*19635 = 18900*19448 = 19040*19305

C(47) = 410810400 = 14364*28600 = 14560*28215 = 14630*28080 = 14820*27720 = 14850*27664 = 15015*27360 = 15048*27300 = 15120*27170 = 15200*27027 = 15400*26676 = 15444*26600 = 15561*26400 = 15600*26334 = 15675*26208 = 15840*25935 = 15960*25740 = 16016*25650 = 16302*25200 = 16380*25080 = 16416*25025 = 16632*24700 = 16720*24570 = 16800*24453 = 17100*24024 = 17160*23940 = 17290*23760 = 17325*23712 = 17550*23408 = 17556*23400 = 17784*23100 = 17955*22880 = 18018*22800 = 18200*22572 = 18480*22230 = 18525*22176 = 18720*21945 = 18810*21840 = 18900*21736 = 19019*21600 = 19152*21450 = 19305*21280 = 19656*20900 = 19760*20790 = 19800*20748 = 19950*20592 = 20020*20520 = 20064*20475

C(43) = 428828400 = 14700*29172 = 14960*28665 = 14994*28600 = 15015*28560 = 15288*28050 = 15300*28028 = 15400*27846 = 15470*27720 = 15600*27489 = 15708*27300 = 15912*26950 = 15925*26928 = 16016*26775 = 16170*26520 = 16380*26180 = 16575*25872 = 16660*25740 = 16830*25480 = 17017*25200 = 17136*25025 = 17160*24990 = 17325*24752 = 17640*24310 = 17680*24255 = 17850*24024 = 18018*23800 = 18200*23562 = 18326*23400 = 18480*23205 = 18564*23100 = 18700*22932 = 19110*22440 = 19404*22100 = 19448*22050 = 19600*21879 = 19635*21840 = 19800*21658 = 19890*21560 = 19992*21450 = 20020*21420 = 20400*21021 = 20475*20944 = 20592*20825

C(44) = 441080640 = 14960*29484 = 15015*29376 = 15120*29172 = 15147*29120 = 15444*28560 = 15470*28512 = 15708*28080 = 15840*27846 = 15912*27720 = 16016*27540 = 16065*27456 = 16320*27027 = 16380*26928 = 16632*26520 = 16830*26208 = 16848*26180 = 17017*25920 = 17136*25740 = 17160*25704 = 17472*25245 = 17680*24948 = 17820*24752 = 17901*24640 = 17952*24570 = 18018*24480 = 18144*24310 = 18360*24024 = 18480*23868 = 18564*23760 = 18720*23562 = 19008*23205 = 19040*23166 = 19278*22880 = 19305*22848 = 19448*22680 = 19635*22464 = 19656*22440 = 19890*22176 = 20020*22032 = 20160*21879 = 20196*21840 = 20592*21420 = 20790*21216 = 20944*21060

C(45) = 490089600 = 15708*31200 = 15840*30940 = 15912*30800 = 16016*30600 = 16320*30030 = 16380*29920 = 16575*29568 = 16800*29172 = 16830*29120 = 17017*28800 = 17136*28600 = 17160*28560 = 17325*28288 = 17472*28050 = 17600*27846 = 17680*27720 = 17850*27456 = 17952*27300 = 18018*27200 = 18200*26928 = 18304*26775 = 18480*26520 = 18564*26400 = 18700*26208 = 18720*26180 = 19040*25740 = 19448*25200 = 19584*25025 = 19635*24960 = 19800*24752 = 19890*24640 = 20020*24480 = 20160*24310 = 20400*24024 = 20475*23936 = 20592*23800 = 20800*23562 = 20944*23400 = 21120*23205 = 21216*23100 = 21420*22880 = 21450*22848 = 21840*22440 = 21879*22400 = 22100*22176

C(46) = 497296800 = 15840*31395 = 15939*31200 = 16016*31050 = 16100*30888 = 16146*30800 = 16380*30360 = 16445*30240 = 16560*30030 = 16632*29900 = 16744*29700 = 16800*29601 = 17160*28980 = 17325*28704 = 17388*28600 = 17550*28336 = 17710*28080 = 17940*27720 = 18018*27600 = 18200*27324 = 18216*27300 = 18400*27027 = 18480*26910 = 18720*26565 = 18837*26400 = 18900*26312 = 18975*26208 = 19305*25760 = 19320*25740 = 19656*25300 = 19734*25200 = 19800*25116 = 19872*25025 = 20020*24840 = 20240*24570 = 20475*24288 = 20592*24150 = 20700*24024 = 20790*23920 = 20930*23760 = 21252*23400 = 21450*23184 = 21528*23100 = 21600*23023 = 21735*22880 = 21840*22770 = 22176*22425

C(51) = 698377680 = 18810*37128 = 19019*36720 = 19152*36465 = 19305*36176 = 19380*36036 = 19448*35910 = 19635*35568 = 19656*35530 = 19760*35343 = 19890*35112 = 20020*34884 = 20196*34580 = 20349*34320 = 20520*34034 = 20592*33915 = 20748*33660 = 20790*33592 = 20944*33345 = 20995*33264 = 21318*32760 = 21420*32604 = 21736*32130 = 21840*31977 = 21879*31920 = 21945*31824 = 22230*31416 = 22440*31122 = 22572*30940 = 22610*30888 = 23205*30096 = 23256*30030 = 23408*29835 = 23562*29640 = 23760*29393 = 23868*29260 = 23940*29172 = 24024*29070 = 24310*28728 = 24453*28560 = 24570*28424 = 24752*28215 = 24871*28080 = 25080*27846 = 25194*27720 = 25245*27664 = 25704*27170 = 25740*27132 = 25840*27027 = 25935*26928 = 26180*26676 = 26334*26520

C(53) = 735134400 = 19305*38080 = 19448*37800 = 19635*37440 = 19656*37400 = 19800*37128 = 19890*36960 = 20020*36720 = 20160*36465 = 20196*36400 = 20400*36036 = 20475*35904 = 20592*35700 = 20790*35360 = 20800*35343 = 20944*35100 = 21216*34650 = 21420*34320 = 21450*34272 = 21600*34034 = 21840*33660 = 21879*33600 = 22100*33264 = 22176*33150 = 22440*32760 = 22464*32725 = 22848*32175 = 22880*32130 = 22950*32032 = 23100*31824 = 23205*31680 = 23400*31416 = 23562*31200 = 23760*30940 = 23800*30888 = 23868*30800 = 24024*30600 = 24310*30240 = 24480*30030 = 24570*29920 = 24640*29835 = 24752*29700 = 25025*29376 = 25200*29172 = 25245*29120 = 25704*28600 = 25740*28560 = 26180*28080 = 26208*28050 = 26400*27846 = 26520*27720 = 26775*27456 = 26928*27300 = 27027*27200

C(52) = 821620800 = 20475*40128 = 20520*40040 = 20592*39900 = 20748*39600 = 20790*39520 = 20800*39501 = 20900*39312 = 21280*38610 = 21450*38304 = 21600*38038 = 21736*37800 = 21840*37620 = 21945*37440 = 22176*37050 = 22230*36960 = 22464*36575 = 22572*36400 = 22800*36036 = 22880*35910 = 23100*35568 = 23400*35112 = 23408*35100 = 23712*34650 = 23760*34580 = 23940*34320 = 24024*34200 = 24453*33600 = 24570*33440 = 24640*33345 = 24700*33264 = 25025*32832 = 25080*32760 = 25200*32604 = 25536*32175 = 25650*32032 = 25740*31920 = 25935*31680 = 26208*31350 = 26334*31200 = 26400*31122 = 26600*30888 = 26676*30800 = 27027*30400 = 27170*30240 = 27300*30096 = 27360*30030 = 27456*29925 = 27664*29700 = 27720*29640 = 28080*29260 = 28215*29120 = 28600*28728

C(50) = 882161280 = 21060*41888 = 21120*41769 = 21216*41580 = 21420*41184 = 21840*40392 = 21879*40320 = 22032*40040 = 22176*39780 = 22440*39312 = 22464*39270 = 22680*38896 = 22848*38610 = 22880*38556 = 23166*38080 = 23205*38016 = 23562*37440 = 23760*37128 = 23868*36960 = 23936*36855 = 24024*36720 = 24192*36465 = 24310*36288 = 24480*36036 = 24570*35904 = 24640*35802 = 24752*35640 = 24948*35360 = 24960*35343 = 25245*34944 = 25704*34320 = 25740*34272 = 25920*34034 = 26180*33696 = 26208*33660 = 26520*33264 = 26928*32760 = 27027*32640 = 27456*32130 = 27540*32032 = 27720*31824 = 27846*31680 = 28080*31416 = 28288*31185 = 28512*30940 = 28560*30888 = 29120*30294 = 29172*30240 = 29376*30030 = 29484*29920 = 29568*29835

C(49) = 908107200 = 21450*42336 = 21560*42120 = 21600*42042 = 21840*41580 = 22050*41184 = 22176*40950 = 22275*40768 = 22464*40425 = 22680*40040 = 22880*39690 = 22932*39600 = 23100*39312 = 23166*39200 = 23400*38808 = 23520*38610 = 23760*38220 = 24024*37800 = 24255*37440 = 24570*36960 = 24640*36855 = 24948*36400 = 25025*36288 = 25200*36036 = 25480*35640 = 25740*35280 = 25872*35100 = 25920*35035 = 26208*34650 = 26325*34496 = 26400*34398 = 26460*34320 = 26950*33696 = 27027*33600 = 27300*33264 = 27456*33075 = 27720*32760 = 28028*32400 = 28080*32340 = 28224*32175 = 28350*32032 = 28512*31850 = 28600*31752 = 28665*31680 = 29106*31200 = 29120*31185 = 29400*30888 = 29484*30800 = 29700*30576 = 30030*30240

C(56) = 1102701600 = 23562*46800 = 23760*46410 = 23800*46332 = 23868*46200 = 24024*45900 = 24310*45360 = 24480*45045 = 24570*44880 = 24752*44550 = 24948*44200 = 25025*44064 = 25200*43758 = 25245*43680 = 25704*42900 = 25740*42840 = 26180*42120 = 26208*42075 = 26325*41888 = 26400*41769 = 26520*41580 = 26775*41184 = 26928*40950 = 27027*40800 = 27300*40392 = 27540*40040 = 27720*39780 = 27846*39600 = 28050*39312 = 28080*39270 = 28350*38896 = 28512*38675 = 28560*38610 = 28600*38556 = 29172*37800 = 29484*37400 = 29700*37128 = 29835*36960 = 29920*36855 = 30030*36720 = 30240*36465 = 30294*36400 = 30600*36036 = 30800*35802 = 30888*35700 = 30940*35640 = 31185*35360 = 31200*35343 = 31416*35100 = 31824*34650 = 32032*34425 = 32130*34320 = 32175*34272 = 32400*34034 = 32725*33696 = 32760*33660 = 33150*33264

C(60) = 1396755360 = 26520*52668 = 26676*52360 = 26928*51870 = 27027*51680 = 27132*51480 = 27170*51408 = 27360*51051 = 27664*50490 = 27720*50388 = 27846*50160 = 28080*49742 = 28215*49504 = 28424*49140 = 28560*48906 = 28728*48620 = 29070*48048 = 29172*47880 = 29260*47736 = 29393*47520 = 29640*47124 = 29835*46816 = 29920*46683 = 30030*46512 = 30096*46410 = 30240*46189 = 30888*45220 = 30940*45144 = 31008*45045 = 31122*44880 = 31416*44460 = 31824*43890 = 31920*43758 = 31977*43680 = 32032*43605 = 32130*43472 = 32604*42840 = 32760*42636 = 33264*41990 = 33345*41888 = 33440*41769 = 33592*41580 = 33660*41496 = 33915*41184 = 34034*41040 = 34272*40755 = 34320*40698 = 34580*40392 = 34884*40040 = 35112*39780 = 35343*39520 = 35360*39501 = 35530*39312 = 35568*39270 = 35910*38896 = 36036*38760 = 36176*38610 = 36465*38304 = 36720*38038 = 36960*37791 = 37128*37620

C(55) = 1491890400 = 27324*54600 = 27600*54054 = 27720*53820 = 28080*53130 = 28336*52650 = 28350*52624 = 28512*52325 = 28600*52164 = 28704*51975 = 28980*51480 = 29484*50600 = 29601*50400 = 29700*50232 = 29808*50050 = 29900*49896 = 30030*49680 = 30240*49335 = 30360*49140 = 30800*48438 = 30888*48300 = 31050*48048 = 31185*47840 = 31200*47817 = 31395*47520 = 31878*46800 = 32032*46575 = 32175*46368 = 32200*46332 = 32292*46200 = 32400*46046 = 32760*45540 = 32890*45360 = 33120*45045 = 33264*44850 = 33488*44550 = 33696*44275 = 34155*43680 = 34320*43470 = 34650*43056 = 34776*42900 = 35100*42504 = 35420*42120 = 35640*41860 = 35880*41580 = 36036*41400 = 36225*41184 = 36400*40986 = 36432*40950 = 36855*40480 = 36960*40365 = 37260*40040 = 37674*39600 = 37800*39468 = 37950*39312 = 38610*38640

C(54) = 1654052400 = 28917*57200 = 29172*56700 = 29484*56100 = 29700*55692 = 29835*55440 = 30030*55080 = 30294*54600 = 30600*54054 = 30800*53703 = 30888*53550 = 30940*53460 = 31185*53040 = 31416*52650 = 31590*52360 = 31824*51975 = 32130*51480 = 32175*51408 = 32400*51051 = 32725*50544 = 32760*50490 = 33048*50050 = 33150*49896 = 33264*49725 = 33660*49140 = 34020*48620 = 34034*48600 = 34320*48195 = 34425*48048 = 34650*47736 = 34749*47600 = 35100*47124 = 35343*46800 = 35640*46410 = 35700*46332 = 35802*46200 = 36036*45900 = 36400*45441 = 36465*45360 = 36720*45045 = 36855*44880 = 37128*44550 = 37400*44226 = 37422*44200 = 37800*43758 = 38556*42900 = 38610*42840 = 38675*42768 = 38896*42525 = 39270*42120 = 39312*42075 = 39600*41769 = 39780*41580 = 40040*41310 = 40392*40950

C(58) = 1715313600 = 29400*58344 = 29920*57330 = 29988*57200 = 30030*57120 = 30576*56100 = 30600*56056 = 30800*55692 = 30940*55440 = 31200*54978 = 31416*54600 = 31680*54145 = 31824*53900 = 31850*53856 = 32032*53550 = 32175*53312 = 32340*53040 = 32487*52800 = 32725*52416 = 32760*52360 = 33150*51744 = 33320*51480 = 33600*51051 = 33660*50960 = 34034*50400 = 34272*50050 = 34320*49980 = 34496*49725 = 34650*49504 = 35035*48960 = 35280*48620 = 35360*48510 = 35700*48048 = 35904*47775 = 36036*47600 = 36400*47124 = 36465*47040 = 36652*46800 = 36960*46410 = 37128*46200 = 37400*45864 = 37440*45815 = 37485*45760 = 38080*45045 = 38220*44880 = 38675*44352 = 38808*44200 = 38896*44100 = 39200*43758 = 39270*43680 = 39600*43316 = 39780*43120 = 39984*42900 = 40040*42840 = 40425*42432 = 40768*42075 = 40800*42042 = 40950*41888 = 41184*41650

C(59) = 1745944200 = 29640*58905 = 29700*58786 = 29835*58520 = 29925*58344 = 30030*58140 = 30600*57057 = 30888*56525 = 30940*56430 = 31122*56100 = 31350*55692 = 31416*55575 = 31977*54600 = 32130*54340 = 32175*54264 = 32300*54054 = 32604*53550 = 32725*53352 = 32760*53295 = 33150*52668 = 33345*52360 = 33592*51975 = 33660*51870 = 33915*51480 = 34034*51300 = 34200*51051 = 34580*50490 = 34650*50388 = 34884*50050 = 35100*49742 = 35112*49725 = 35343*49400 = 35530*49140 = 35700*48906 = 35910*48620 = 36036*48450 = 36465*47880 = 36575*47736 = 37050*47124 = 37128*47025 = 37400*46683 = 37620*46410 = 37791*46200 = 37800*46189 = 38038*45900 = 38610*45220 = 38675*45144 = 38760*45045 = 39270*44460 = 39501*44200 = 39780*43890 = 39900*43758 = 40040*43605 = 40392*43225 = 40698*42900 = 40755*42840 = 40950*42636 = 41496*42075 = 41580*41990 = 41769*41800

Going deeper would require more ram or counting one subinterval at a time. PrimeHunter (talk) 18:55, 15 June 2012 (UTC)[reply]

Existence of the Inverse Laplace transform

Hello, I want to know what is the condition that the function $F(p]$ should satisfy to say that its Inverse Laplace transform exist. I want to prove that the Inverse Laplce Transform of the function $F(p)=\cos(p)$ doesn't exist and I don't know how to begin. Thank you Rabah201130 (talk) 07:01, 12 June 2012 (UTC)[reply]

As I recall, the inverse Laplace transform is defined to exist for a function if and only if the improper integral converges. So, show that:

{\mathcal {L}}^{-1}cos(p)={\frac {1}{2\pi i}}\int _{-i\infty }^{+i\infty }e^{st}cos(s)\,ds

...does not converge. If you're pretty good at integral calculus, you can probably see how to show this directly for cosine.

For a more general case, showing the existence of the inverse Laplace transform can be much more difficult. There's a lot of symmetry between a Laplace transform and its inverse, so the techniques described in our section on convergence of the forward Laplace integral introduce useful concepts - including a lot of measure theory tools - that will help you formally explore additional constraints and necessary conditions for the existence of a region of convergence for a Laplace integral, given a particular function. Nimur (talk) 19:15, 12 June 2012 (UTC)[reply]

Are the integration limits not supposed to have a factor

i

? — Quondum ^☏ 20:46, 12 June 2012 (UTC)[reply]

My error, you are correct. I originally forgot the complex limits when transcribing the MathML. I have fixed the equation above. Nimur (talk) 02:20, 13 June 2012 (UTC)[reply]

Mathematical Complex Mapping

I am extremely new with Mathematica, so a simple explanation or just some code that I could start off with would be extremely helpful. Basically, I want to take an image, and then apply a complex conformal map to it, so that the image stretches and distorts just as we usually represent with a grid being stretched and distorted in concordance with a conformal map. I feel like this would be easier to visualize, and may give some insight into sets such as the Julia fractal. — Trevor K. — 17:25, 12 June 2012 (UTC) — Preceding unsigned comment added by Yakeyglee (talk • contribs)

It depends somewhat on what version you're using. If it's 8, you can fetch (interpolated) pixel values with ImageValue; if it's 7, you can fetch them all at once with ImageData and then use ListInterpolation; if it's earlier, see the basic how-to for fetching the data. The rest of that last article is relevant regardless of version: it shows how to import/export images and how to construct transformed images. The rest is something like

DensityPlot[With[{z=f[x+I y]},g[Re[z],Im[z]]],{x,-1,1},{y,-1,1},Mesh->False]

where f is your conformal map and g is ImageValue or the result of ListInterpolation. --Tardis (talk) 03:36, 13 June 2012 (UTC)[reply]

The exterior product: inconsistent definitions?

In trying to find a definition of the exterior (or wedge) product of the exterior algebra in terms of the tensor product, I am having difficulty reconciling different definitions, specifically with regard to a multiplicative constant. My own understanding of the formal definitions is a little sketchy, so I'd appreciate a little elucidation from others who may be more familiar with the topic.

There seem to me to be two non-equivalent definitions:

Definition 1: As a quotient algebra. In this definition I expect all properties to be inherited.

Definition 2: As the fully antisymmetric subspace of the tensor algebra T(V*), forming an algebra with the operations of addition (+) and multiplication (∧), where addition (+) is as for the tensor algebra, and multiplication (∧) is defined as a suitable extension of the definition on 1-forms (covectors) as a ∧ b = a ⊗ b − b ⊗ a. This appears to be pretty common in the literature.

In each case, it would seem that a consistent algebra is created, but that the second case does not have the property of inheritance; it would only be achieved if we defined a ∧ b = (a ⊗ b − b ⊗ a)/2. A property of significance is that of a 2-form acting on two vectors, e.g.:

ω = e¹ ⊗ e² − e² ⊗ e¹

⇒ ω(e₁, e₂) = e¹(e₁)e²(e₂) − e²(e₁)e¹(e₂) = 1

but

ω = e¹ ∧ e² − e² ∧ e¹ = (e¹ ⊗ e² − e² ⊗ e¹) − (e² ⊗ e¹ − e¹ ⊗ e²) = 2e¹ ⊗ e² − 2e² ⊗ e¹

⇒ ω(e₁, e₂) = 2e¹(e₁)e²(e₂) − 2e²(e₁)e¹(e₂) = 2

Am I correct in saying these definitions are non-equivalent? If so, what is the modern perspective? — Quondum ^☏ 20:20, 12 June 2012 (UTC)[reply]

There's no contradiction between the two definitions as stated. Defining the exterior algebra as a quotient doesn't determine what equivalence class "a∧b" should denote. We still have to define what that symbol means. We could hypothetically define it either way you mention, but the usual definition is that a∧b = a⊗b − b⊗a. This ensures that (e¹∧e²)(e₁,e₂) = 1, which is nice for geometric reasons. For example if V has dimension n, then (e¹∧...∧eⁿ)(v₁,...,v_n) measures the volume of the parallelpiped formed by v₁,...,v_n. Maybe you think a more natural definition would be to have e¹∧e² denote the equivalence class of e¹⊗e², but that's not the convention at least in the literature I've seen. Rckrone (talk) 06:13, 13 June 2012 (UTC)[reply]

I think we're agreed that the two definitions are equivalent as exterior algebras prior to incorporating the action of p-forms on vectors. However, the addition of this action in terms of the action in the parent tensor algebra does seem to introduce a distinction (they're not isomorphic on this operation due to the scalar multiplier). I am doubtful of the "nice for geometric reasons" that you mention, because geometric algebra fits definition 1 rather than definition 2, and it provides a clean geometric interpretation of the exterior product and many other constructions, without the introduction of awkward multipliers of p!; it has that v₁∧...∧v_p is a direct representation of the oriented p-volume of a parallelotope in n-D space (p≤n), which does not need the (IMO somewhat arbitrary) multilinear function you give for this.

As an aside, the article exterior algebra seems to use both definitions in different parts of it, without highlighting the distinction. For example, in exterior algebra#Universal property, definition 1 is used, which would imply for any element of the exterior algebra that

\omega \wedge \eta =\operatorname {Alt} (\omega \otimes \eta )

Whereas in exterior algebra#Alternating multilinear forms is stated for a k-form and an m-form that

\omega \wedge \eta ={\frac {(k+m)!}{k!\,m!}}\operatorname {Alt} (\omega \otimes \eta )

Or, as is certainly possible, have I got something wrong? I am primarily concerned that while the choice is probably arbitrary, confusing the two approaches is problematic. — Quondum ^☏ 09:53, 13 June 2012 (UTC)[reply]

No I think you're right. And as far as Alt is concerned I'd prefer if the version there with the 1/n! factor were called the antisymmetrization rather than the alternating product. And yesthe version giving the volume is use in mathematics but for some reason I don't understand the antisymmetrization version seems to have also become popular from quantum mechanics. If the version of Alt meaning antisymmetrization were used then the volume form would be n! Alt, the two versions you've got are okay because they use two different Alt's! It would really help if there were good citations in the article so the confusion could be delineated and controlled. Dmcq (talk) 08:56, 17 June 2012 (UTC)[reply]