I'm searching for some very big {more less $2^{1024}$} highly composite number (HCN), for a number theory experiment, so I asked myself "if there are prime list this size, why not HCN lists this size" so instead of searching them, I could just copy them, but I have not found any list enough big, I have just found this list: https://oeis.org/A002182/list but is verry small.
I could work with a not so big list, but I still need lots of very big, very composite numbers
You can compute your (probably) own, if that is good enough. If you look at the divisor function the number of divisors of a number $N$ that is $p^aq^br^c \ldots$ (with $p,q,r$ prime) is $(a+1)(b+1)(c+1)\ldots$. Clearly you want $p=2, q=3, r=5$ and $a \ge b \ge c$ and so on. Now if you increase $a$ by $1$, you multiply the number of factors by $\frac {a+2}{a+1}=1+\frac 1{a+1}$ If we consider $\Delta \frac {\log \sigma_0(N)}{\Delta \log N}$ for increasing the exponent on a prime $P$ that currently has exponent $A$, we get $ \frac {\Delta\log \sigma_0(N)}{\Delta \log N}\approx \frac{\frac 1{A+1}}{\log P}$ This expresses the ratio of the (log of) the additional factors we get to the increase in the (log of) $N$. To have a highly composite number we want these ratios to be about the same. They can't be absolutely the same because of the granularity of the primes and exponents. You can pick a $k=\frac{\frac 1{A+1}}{\log P}$ and find $A=\frac 1{k \log P}-1$ If you pick $k=0.2$ and round all the numbers to the nearest integer, you get $2^6*3^4*5^2*7^2*11*13*17*19*23=6746328388800$, which is number $104$ on the list. As you pick smaller positive $k$ you get bigger numbers. Because of the granularity of integers, there will be other candidates. If you really need the record setters, you should check around nearby exponents to find if the one you started with is really highly composite.
Here is a list of the first 10000 ones.
I found this in the External Links section of the highly composite number Wikipedia page, which has at least one other link that you might find useful.
Alright, had this; here are about 65 SHC numbers larger than $2^{996}$ I had the program put a blank line before the number if it is double the previous. The first four are below $2^{1024},$ the others are larger, and get much larger...
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15560475689982524871754948583806252766855067946138645224878749732945057767066914899980926728598563187341147403918283422933332145030246689502024323459068376813099506555771482080033527942678972445339259799009616293851768869720270204625491367242118535891543566704711937327085276281210304794752955524782237463933076405146789038884520468155187239101953419356211030087578282441057551991538505137038593465184276948842655400160947939035374316614007155474659915064038400000
16073971387751948192522861887071859108161285188361220517299748474132244673380123091680297310642315772523405268247586775890132105816244830255591126133217633247931790272111940988674634364787378536035455372376933631548877242421039121378132582361108447575964504405967431258879090398490244852979803057100051300242867926516633077167709643604308417992317882194965994080468365761612451207259275806560867049535358088154463028366259221023541669062269391605323692261151667200000
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17519198229196113606460784922200377032435174500931966215230686159189948924208403338976364522139477025946757159320995550497189773582289219188701579874981363870886585467267797714907359415159776527588939200362716111295068344164357169820361860973778069205967048961612368970796168295308901257956070129766974012699004424657670073768935585370415392162425175300997581574237356995604788307755164553604561166826131669218518953709697953844989324069327094873949950756044074749619200000
18412677338885115400390284953232596261089368400479496492207451153308636319343031909264159112768590354270041774446366323572546452034985969367325360448605413428301801326098455398367634745332925130495975099581214632971116829716739385481200315883440750735471368458654599788306772878369655222111829706385089687346653650315211247531151300224306577162708859241348458234523462202380632511450677945838393786334264384348663420348892549491083779596862776712521398244602322561849779200000
36825354677770230800780569906465192522178736800958992984414902306617272638686063818528318225537180708540083548892732647145092904069971938734650720897210826856603602652196910796735269490665850260991950199162429265942233659433478770962400631766881501470942736917309199576613545756739310444223659412770179374693307300630422495062302600448613154325417718482696916469046924404761265022901355891676787572668528768697326840697785098982167559193725553425042796489204645123699558400000
39071701313114214879628184670759569266031639745817491556464211347320926269645913711458545637294948731761028645375189338620943571218240226997464414871940687294856422413980922355336120929596467126912459161311337451164709912658920975991107070304661273060670243869265060750786972047900408381321302636949160316549599045968878267261103059075978556739268199310141428373658786793451702189298338601069071614601309023587863777980349990020079780304542812183970407075046128476245231462400000
If what you want is, say, just one really enormous number that is highly composite, the least work is https://en.wikipedia.org/wiki/Superior_highly_composite_number
https://oeis.org/A002201/b002201.txt
Note that entry 150 is about $2^{953}.$ It would be necessary to extend just a few more entries to reach $2^{1024}$
I have programmed SHC numbers, it is not a task for beginners, but is possible.
Guy Robin found a method for interpolating between consecutive Superior Highly Composite numbers, resulting in all the highly composite numbers between. The original is in French. Discussion in http://math.univ-lyon1.fr/~nicolas/hcnrevisited.pdf
The reason for mentioning this is that you will need to compute such an enormous list yourself.
Simplified the two programs, which previously allowed many, many choices....
jagy@phobeusjunior:~$
jagy@phobeusjunior:~$ date
Tue Jan 3 10:22:20 PST 2017
jagy@phobeusjunior:~$
jagy@phobeusjunior:~$ ./Superior_Highly_Composite_construct_deltas 0.2 > temp.txt
jagy@phobeusjunior:~$ cat temp.txt
1.0000000000000000 2 1
0.5849625007211562 2 2
0.4150374992788437 2 3
0.3219280948873623 2 4
0.2630344058337938 2 5
0.2223924213364480 2 6
0.6309297535714574 3 1
0.3690702464285426 3 2
0.2618595071429148 3 3
0.2031140135750123 3 4
0.4306765580733931 5 1
0.2519296364125923 5 2
0.3562071871080222 7 1
0.2083678469455574 7 2
0.2890648263178878 11 1
0.2702381544273197 13 1
0.2446505421182260 17 1
0.2354089133666382 19 1
0.2210647294575037 23 1
0.2058468324604344 29 1
0.2018490865820999 31 1
jagy@phobeusjunior:~$ sort -n -r temp.txt > Superior_Highly_Composite_Deltas.txt
jagy@phobeusjunior:~$ cat Superior_Highly_Composite_Deltas.txt
1.0000000000000000 2 1
0.6309297535714574 3 1
0.5849625007211562 2 2
0.4306765580733931 5 1
0.4150374992788437 2 3
0.3690702464285426 3 2
0.3562071871080222 7 1
0.3219280948873623 2 4
0.2890648263178878 11 1
0.2702381544273197 13 1
0.2630344058337938 2 5
0.2618595071429148 3 3
0.2519296364125923 5 2
0.2446505421182260 17 1
0.2354089133666382 19 1
0.2223924213364480 2 6
0.2210647294575037 23 1
0.2083678469455574 7 2
0.2058468324604344 29 1
0.2031140135750123 3 4
0.2018490865820999 31 1
jagy@phobeusjunior:~$ ./Superior_Highly_Composite_read > Superior_Highly_Composite_read.txt
jagy@phobeusjunior:~$ cat Superior_Highly_Composite_read.txt
2
6
12
60
120
360
2520
5040
55440
720720
1441440
4324320
21621600
367567200
6983776800
13967553600
321253732800
2248776129600
65214507758400
195643523275200
6064949221531200
jagy@phobeusjunior:~$
jagy@phobeusjunior:~$ date
Tue Jan 3 10:23:39 PST 2017
=================================================================
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
using namespace std;
// g++ -o Superior_Highly_Composite_construct_deltas Superior_Highly_Composite_construct_deltas.cc -lgmp -lgmpxx
// ./Superior_Highly_Composite_construct_deltas
// save in temp.txt then
// sort -n -r temp.txt > Superior_Highly_Composite_Deltas.txt
int PrimeQ(int i)
{
if ( i < 0 ) i *= -1;
if ( i <= 3) return 1;
else
{
int boo = 1;
int j = 2;
while (boo && j * j <= i )
{
if ( i % j == 0) boo = 0;
++j;
}
return boo;
}
}
int main(int argc, char *argv[])
{
if ( argc != 2) cout << "Usage: ./Superior_Highly_Composite_construct_deltas LowerBound " << endl;
else {
double bound = 0.5 ;
bound = atof(argv[1]);
for (int p = 2; p < 100100; ++p)
{
if( PrimeQ(p) )
{
double eps = 1.0;
for (int k = 1; k <= 500 && eps > bound; ++k)
{
double mu = (1.0 + k) / (1.0 * k) ;
mu = log(mu);
mu /= log(1.0 * p);
cout.precision(16);
cout.setf(ios::fixed, ios::floatfield); // DD page 582
if ( mu > bound) cout << mu << setw(12) << p << setw(12) << k << endl;
eps = mu;
} // for k
} // if prime
}// for p
} // else argc
return 0 ;
}
=================================================================
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
using namespace std;
const int SIZE = 80;
// g++ -o Superior_Highly_Composite_read Superior_Highly_Composite_read.cc -lgmp -lgmpxx
// ./Superior_Highly_Composite_read
int main()
{
char line[SIZE];
ifstream ifs("Superior_Highly_Composite_Deltas.txt", ios::in ); // DD page 659
if (!ifs)
{
cerr << "could not open file for reading " << endl;
exit(1);
}
int pp, kk;
int shc_count = 0;
mpz_class n, sigma;
n = 1; sigma = 1;
double delta_new;
while(ifs.getline( line,SIZE) && !ifs.eof() ) // DD page 661
{
istrstream lineBuffer(line, SIZE); // DD page 678
lineBuffer >> delta_new >> pp >> kk ;
shc_count++ ;
// if ( pp == 2 ) cout << endl;
n *= pp;
cout << n << endl;
} // while not eof
return 0 ;
}
=================================================================