[Math] Smallest number with at least $n$ divisors.

Question

I have seen lots of posts on "exactly $n$ divisors" and understood the process as well, but I can't seem to find or come up with an algorithm, apart from brute force, for "at least $n$ divisors".

Since supposes we are looking for a number with number of divisors > 10;
11 is the first number that's > 10, but the smallest number that has 11 divisors (including 1 and itself) is 2^10 i.e 1024….
while if we consider 12, [12 is also > 10]… the smallest number that has 12 divisors is 60 , which is way less than 1024.

Is there an efficient algo apart from just looping and checking the num of divisors till we find a num whose #divisors > n ?

Answer 1

I am addressing the question of finding the smallest number with AT LEAST $N$ divisors. Trying to get exactly $N$ divisors is a different level of nightmare.

well, i cannot tell how much work you are willing to put into this. Guy Robin came up with a method for finding the highly composite numbers between two consecutitive superior highly composite numbers.

Assuming that is too ambitious, do this: first find the first primorial number $P =2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdots p_k$ that has more divisors than you need. That would mean $2^k$ is larger than you want.

Next, make a fairly complicated multiple loop to produce all the numbers $$ 2^{e_2} 3^{e_3} 5^{e_5} \cdots p_k^{e_{p_k}} \; \leq \; \; P $$ WITH $$ e_2 \geq e_3 \geq e_5 \geq \ldots \geq e_{p_k} \geq 0 $$ then calculate the number of divisors ( depends only on the exponents) and calculate the number itself.

The winner, smallest such number with at least some $N$ divisors, is likely to have a number of exponents zero at the end. The general pattern is that highly composite numbers are fairly similar to the least common multiple $L$ of $1,2,3,4,5,6,7,\ldots, w$ as far as the relative sizes of the prime exponents when the whole thing is factored. I should point out that the prime number theorem says $\log L \approx W.$

A little more efficiency: we can combine the two bounds. We can find the smallest LCM $L$ that has enough divisors. This will give a better (smaller) upper bound. Then, using the set of primes we used in producing the primorial $P,$ we can produce all the numbers $$ 2^{e_2} 3^{e_3} 5^{e_5} \cdots p_k^{e_{p_k}} \; \leq \; \; L $$ WITH $$ e_2 \geq e_3 \geq e_5 \geq \ldots \geq e_{p_k} \geq 0 $$

===================================

Fri Apr 27 11:08:30 PDT 2018
Fri Apr 27 11:08:56 PDT 2018
       d(n)    n    factored
         1     1  =    1 
         2     2  =   2
         3     4  =   2^2
         4     6  =   2 3
         4     8  =   2^3
         5     16  =   2^4
         6     12  =   2^2 3
         6     32  =   2^5
         7     64  =   2^6
         8     128  =   2^7
         8     24  =   2^3 3
         8     30  =   2 3 5
         9     256  =   2^8
         9     36  =   2^2 3^2
        10     48  =   2^4 3
        10     512  =   2^9
        11     1024  =   2^10
        12     2048  =   2^11
        12     60  =   2^2 3 5
        12     72  =   2^3 3^2
        12     96  =   2^5 3
        13     4096  =   2^12
        14     192  =   2^6 3
        14     8192  =   2^13
        15     144  =   2^4 3^2
        15     16384  =   2^14
        16     120  =   2^3 3 5
        16     210  =   2 3 5 7
        16     216  =   2^3 3^3
        16     32768  =   2^15
        16     384  =   2^7 3
        17     65536  =   2^16
        18     131072  =   2^17
        18     180  =   2^2 3^2 5
        18     288  =   2^5 3^2
        18     768  =   2^8 3
        19     262144  =   2^18
        20     1536  =   2^9 3
        20     240  =   2^4 3 5
        20     432  =   2^4 3^3
        20     524288  =   2^19
        21     1048576  =   2^20
        21     576  =   2^6 3^2
        22     2097152  =   2^21
        22     3072  =   2^10 3
        23     4194304  =   2^22
        24     1152  =   2^7 3^2
        24     360  =   2^3 3^2 5
        24     420  =   2^2 3 5 7
        24     480  =   2^5 3 5
        24     6144  =   2^11 3
        24     8388608  =   2^23
        24     864  =   2^5 3^3
        25     1296  =   2^4 3^4
        26     12288  =   2^12 3
        27     2304  =   2^8 3^2
        27     900  =   2^2 3^2 5^2
        28     1728  =   2^6 3^3
        28     24576  =   2^13 3
        28     960  =   2^6 3 5
        30     2592  =   2^5 3^4
        30     4608  =   2^9 3^2
        30     49152  =   2^14 3
        30     720  =   2^4 3^2 5
        32     1080  =   2^3 3^3 5
        32     1920  =   2^7 3 5
        32     2310  =   2 3 5 7 11
        32     3456  =   2^7 3^3
        32     840  =   2^3 3 5 7
        32     98304  =   2^15 3
        33     9216  =   2^10 3^2
        34     196608  =   2^16 3
        35     5184  =   2^6 3^4
        36     1260  =   2^2 3^2 5 7
        36     1440  =   2^5 3^2 5
        36     1800  =   2^3 3^2 5^2
        36     18432  =   2^11 3^2
        36     3840  =   2^8 3 5
        36     393216  =   2^17 3
        36     6912  =   2^8 3^3
        36     7776  =   2^5 3^5
        38     786432  =   2^18 3
        39     36864  =   2^12 3^2
        40     10368  =   2^7 3^4
        40     13824  =   2^9 3^3
        40     1572864  =   2^19 3
        40     1680  =   2^4 3 5 7
        40     2160  =   2^4 3^3 5
        40     7680  =   2^9 3 5
        42     15552  =   2^6 3^5
        42     2880  =   2^6 3^2 5
        42     3145728  =   2^20 3
        42     73728  =   2^13 3^2
        44     15360  =   2^10 3 5
        44     27648  =   2^10 3^3
        44     6291456  =   2^21 3
        45     147456  =   2^14 3^2
        45     20736  =   2^8 3^4
        45     3600  =   2^4 3^2 5^2
        48     2520  =   2^3 3^2 5 7
        48     294912  =   2^15 3^2
        48     30720  =   2^11 3 5
        48     31104  =   2^7 3^5
        48     3360  =   2^5 3 5 7
        48     4320  =   2^5 3^3 5
        48     4620  =   2^2 3 5 7 11
        48     5400  =   2^3 3^3 5^2
        48     55296  =   2^11 3^3
        48     5760  =   2^7 3^2 5
        49     46656  =   2^6 3^6
        50     41472  =   2^9 3^4
        50     6480  =   2^4 3^4 5
        51     589824  =   2^16 3^2
        52     110592  =   2^12 3^3
        52     61440  =   2^12 3 5
        54     11520  =   2^8 3^2 5
        54     1179648  =   2^17 3^2
        54     62208  =   2^8 3^5
        54     6300  =   2^2 3^2 5^2 7
        54     7200  =   2^5 3^2 5^2
        55     82944  =   2^10 3^4
        56     122880  =   2^13 3 5
        56     221184  =   2^13 3^3
        56     6720  =   2^6 3 5 7
        56     8640  =   2^6 3^3 5
        56     93312  =   2^7 3^6
        57     2359296  =   2^18 3^2
        60     10800  =   2^4 3^3 5^2
        60     124416  =   2^9 3^5
        60     12960  =   2^5 3^4 5
        60     165888  =   2^11 3^4
        60     23040  =   2^9 3^2 5
        60     245760  =   2^14 3 5
        60     442368  =   2^14 3^3
        60     4718592  =   2^19 3^2
        60     5040  =   2^4 3^2 5 7
        63     14400  =   2^6 3^2 5^2
        63     186624  =   2^8 3^6
        63     9437184  =   2^20 3^2
        64     13440  =   2^7 3 5 7
        64     17280  =   2^7 3^3 5
        64     27000  =   2^3 3^3 5^3
        64     279936  =   2^7 3^7
        64     30030  =   2 3 5 7 11 13
        64     491520  =   2^15 3 5
        64     7560  =   2^3 3^3 5 7
        64     884736  =   2^15 3^3
        64     9240  =   2^3 3 5 7 11
        65     331776  =   2^12 3^4
        66     248832  =   2^10 3^5
        66     46080  =   2^10 3^2 5
        68     1769472  =   2^16 3^3
        68     983040  =   2^16 3 5
        70     25920  =   2^6 3^4 5
        70     373248  =   2^9 3^6
        70     663552  =   2^13 3^4
        72     10080  =   2^5 3^2 5 7
        72     12600  =   2^3 3^2 5^2 7
        72     13860  =   2^2 3^2 5 7 11
        72     1966080  =   2^17 3 5
        72     21600  =   2^5 3^3 5^2
        72     26880  =   2^8 3 5 7
        72     28800  =   2^7 3^2 5^2
        72     34560  =   2^8 3^3 5
        72     3538944  =   2^17 3^3
        72     38880  =   2^5 3^5 5
        72     497664  =   2^11 3^5
        72     559872  =   2^8 3^7
        72     92160  =   2^11 3^2 5
        75     1327104  =   2^14 3^4
        75     32400  =   2^4 3^4 5^2
        76     3932160  =   2^18 3 5
        76     7077888  =   2^18 3^3
        77     746496  =   2^10 3^6
        78     184320  =   2^12 3^2 5
        78     995328  =   2^12 3^5
        80     1119744  =   2^9 3^7
        80     15120  =   2^4 3^3 5 7
        80     18480  =   2^4 3 5 7 11
        80     2654208  =   2^15 3^4
        80     51840  =   2^7 3^4 5
        80     53760  =   2^9 3 5 7
        80     54000  =   2^4 3^3 5^3
        80     69120  =   2^9 3^3 5
        80     7864320  =   2^19 3 5
        81     1679616  =   2^8 3^8
        81     44100  =   2^2 3^2 5^2 7^2
        81     57600  =   2^8 3^2 5^2
        84     1492992  =   2^11 3^6
        84     1990656  =   2^13 3^5
        84     20160  =   2^6 3^2 5 7
        84     368640  =   2^13 3^2 5
        84     43200  =   2^6 3^3 5^2
        84     77760  =   2^6 3^5 5
        85     5308416  =   2^16 3^4
        88     107520  =   2^10 3 5 7
        88     138240  =   2^10 3^3 5
        88     2239488  =   2^10 3^7
        90     103680  =   2^8 3^4 5
        90     115200  =   2^9 3^2 5^2
        90     25200  =   2^4 3^2 5^2 7
        90     3359232  =   2^9 3^8
        90     3981312  =   2^14 3^5
        90     64800  =   2^5 3^4 5^2
        90     737280  =   2^14 3^2 5
        91     2985984  =   2^12 3^6
        96     108000  =   2^5 3^3 5^3
        96     1474560  =   2^15 3^2 5
        96     155520  =   2^7 3^5 5
        96     215040  =   2^11 3 5 7
        96     276480  =   2^11 3^3 5
        96     27720  =   2^3 3^2 5 7 11
        96     30240  =   2^5 3^3 5 7
        96     36960  =   2^5 3 5 7 11
        96     37800  =   2^3 3^3 5^2 7
        96     40320  =   2^7 3^2 5 7
        96     4478976  =   2^11 3^7
        96     60060  =   2^2 3 5 7 11 13
        96     7962624  =   2^15 3^5
        96     86400  =   2^7 3^3 5^2
        98     233280  =   2^6 3^6 5
        98     5971968  =   2^13 3^6
        99     230400  =   2^10 3^2 5^2
        99     6718464  =   2^10 3^8
       100     162000  =   2^4 3^4 5^3
       100     207360  =   2^9 3^4 5
       100     45360  =   2^4 3^4 5 7
       102     2949120  =   2^16 3^2 5
       104     430080  =   2^12 3 5 7
       104     552960  =   2^12 3^3 5
       104     8957952  =   2^12 3^7
       105     129600  =   2^6 3^4 5^2
       108     172800  =   2^8 3^3 5^2
       108     194400  =   2^5 3^5 5^2
       108     311040  =   2^8 3^5 5
       108     460800  =   2^11 3^2 5^2
       108     50400  =   2^5 3^2 5^2 7
       108     5898240  =   2^17 3^2 5
       108     69300  =   2^2 3^2 5^2 7 11
       108     80640  =   2^8 3^2 5 7
       108     88200  =   2^3 3^2 5^2 7^2
       110     414720  =   2^10 3^4 5
       112     1105920  =   2^13 3^3 5
       112     216000  =   2^6 3^3 5^3
       112     466560  =   2^7 3^6 5
       112     60480  =   2^6 3^3 5 7
       112     73920  =   2^6 3 5 7 11
       112     860160  =   2^13 3 5 7
       117     921600  =   2^12 3^2 5^2
       120     161280  =   2^9 3^2 5 7
       120     1720320  =   2^14 3 5 7
       120     2211840  =   2^14 3^3 5
       120     259200  =   2^7 3^4 5^2
       120     324000  =   2^5 3^4 5^3
       120     345600  =   2^9 3^3 5^2
       120     55440  =   2^4 3^2 5 7 11
       120     622080  =   2^9 3^5 5
       120     75600  =   2^4 3^3 5^2 7
       120     829440  =   2^11 3^4 5
       120     90720  =   2^5 3^4 5 7
       125     810000  =   2^4 3^4 5^4
       126     100800  =   2^6 3^2 5^2 7
       126     1843200  =   2^13 3^2 5^2
       126     388800  =   2^6 3^5 5^2
       126     933120  =   2^8 3^6 5
       128     120120  =   2^3 3 5 7 11 13
       128     120960  =   2^7 3^3 5 7
       128     1399680  =   2^7 3^7 5
       128     147840  =   2^7 3 5 7 11
       128     189000  =   2^3 3^3 5^3 7
       128     3440640  =   2^15 3 5 7
       128     432000  =   2^7 3^3 5^3
       128     4423680  =   2^15 3^3 5
       128     510510  =   2 3 5 7 11 13 17
       128     83160  =   2^3 3^3 5 7 11
       130     1658880  =   2^12 3^4 5
       132     1244160  =   2^10 3^5 5
       132     322560  =   2^10 3^2 5 7
       132     691200  =   2^10 3^3 5^2
       135     176400  =   2^4 3^2 5^2 7^2
       135     3686400  =   2^14 3^2 5^2
       135     518400  =   2^8 3^4 5^2
       136     6881280  =   2^16 3 5 7
       136     8847360  =   2^16 3^3 5
       140     181440  =   2^6 3^4 5 7
       140     1866240  =   2^9 3^6 5
       140     3317760  =   2^13 3^4 5
       140     648000  =   2^6 3^4 5^3
       144     110880  =   2^5 3^2 5 7 11
       144     1382400  =   2^11 3^3 5^2
       144     138600  =   2^3 3^2 5^2 7 11
       144     151200  =   2^5 3^3 5^2 7
       144     180180  =   2^2 3^2 5 7 11 13
       144     201600  =   2^7 3^2 5^2 7
       144     241920  =   2^8 3^3 5 7
       144     2488320  =   2^11 3^5 5
       144     264600  =   2^3 3^3 5^2 7^2
       144     272160  =   2^5 3^5 5 7
       144     2799360  =   2^8 3^7 5
       144     295680  =   2^8 3 5 7 11
       144     645120  =   2^11 3^2 5 7
       144     7372800  =   2^15 3^2 5^2
       144     777600  =   2^7 3^5 5^2
       144     864000  =   2^8 3^3 5^3
       144     972000  =   2^5 3^5 5^3
       147     1166400  =   2^6 3^6 5^2
       150     1036800  =   2^9 3^4 5^2
       150     1620000  =   2^5 3^4 5^4
       150     226800  =   2^4 3^4 5^2 7
       150     6635520  =   2^14 3^4 5
       154     3732480  =   2^10 3^6 5
       156     1290240  =   2^12 3^2 5 7
       156     2764800  =   2^12 3^3 5^2
       156     4976640  =   2^12 3^5 5
       160     1296000  =   2^7 3^4 5^3
       160     166320  =   2^4 3^3 5 7 11
       160     1728000  =   2^9 3^3 5^3
       160     240240  =   2^4 3 5 7 11 13
       160     362880  =   2^7 3^4 5 7
       160     378000  =   2^4 3^3 5^3 7
       160     483840  =   2^9 3^3 5 7
       160     5598720  =   2^9 3^7 5
       160     591360  =   2^9 3 5 7 11
       162     1555200  =   2^8 3^5 5^2
       162     352800  =   2^5 3^2 5^2 7^2
       162     403200  =   2^8 3^2 5^2 7
       162     485100  =   2^2 3^2 5^2 7^2 11
       162     8398080  =   2^8 3^8 5
       165     2073600  =   2^10 3^4 5^2
       168     1944000  =   2^6 3^5 5^3
       168     221760  =   2^6 3^2 5 7 11
       168     2332800  =   2^7 3^6 5^2
       168     2580480  =   2^13 3^2 5 7
       168     302400  =   2^6 3^3 5^2 7
       168     544320  =   2^6 3^5 5 7
       168     5529600  =   2^13 3^3 5^2
       168     7464960  =   2^11 3^6 5
       168     9953280  =   2^13 3^5 5
       175     3240000  =   2^6 3^4 5^4
       176     1182720  =   2^10 3 5 7 11
       176     3456000  =   2^10 3^3 5^3
       176     967680  =   2^10 3^3 5 7
       180     2592000  =   2^8 3^4 5^3
       180     277200  =   2^4 3^2 5^2 7 11
       180     3110400  =   2^9 3^5 5^2
       180     4147200  =   2^11 3^4 5^2
       180     453600  =   2^5 3^4 5^2 7
       180     4860000  =   2^5 3^5 5^4
       180     5160960  =   2^14 3^2 5 7
       180     529200  =   2^4 3^3 5^2 7^2
       180     725760  =   2^8 3^4 5 7
       180     806400  =   2^9 3^2 5^2 7
       189     4665600  =   2^8 3^6 5^2
       189     705600  =   2^6 3^2 5^2 7^2
       192     1021020  =   2^2 3 5 7 11 13 17
       192     1088640  =   2^7 3^5 5 7
       192     1323000  =   2^3 3^3 5^3 7^2
       192     1935360  =   2^11 3^3 5 7
       192     2365440  =   2^11 3 5 7 11
       192     332640  =   2^5 3^3 5 7 11
       192     360360  =   2^3 3^2 5 7 11 13
       192     3888000  =   2^7 3^5 5^3
       192     415800  =   2^3 3^3 5^2 7 11
       192     443520  =   2^7 3^2 5 7 11
       192     480480  =   2^5 3 5 7 11 13
       192     604800  =   2^7 3^3 5^2 7
       192     6912000  =   2^11 3^3 5^3
       192     6998400  =   2^7 3^7 5^2
       192     756000  =   2^5 3^3 5^3 7
       195     8294400  =   2^12 3^4 5^2
       196     1632960  =   2^6 3^6 5 7
       196     5832000  =   2^6 3^6 5^3
       198     1612800  =   2^10 3^2 5^2 7
       198     6220800  =   2^10 3^5 5^2
       200     1134000  =   2^4 3^4 5^3 7
       200     1451520  =   2^9 3^4 5 7
       200     498960  =   2^4 3^4 5 7 11
       200     5184000  =   2^9 3^4 5^3
       200     6480000  =   2^7 3^4 5^4
       208     3870720  =   2^12 3^3 5 7
       208     4730880  =   2^12 3 5 7 11
       210     907200  =   2^6 3^4 5^2 7
       210     9331200  =   2^9 3^6 5^2
       210     9720000  =   2^6 3^5 5^4
       216     1058400  =   2^5 3^3 5^2 7^2
       216     1209600  =   2^8 3^3 5^2 7
       216     1360800  =   2^5 3^5 5^2 7
       216     1411200  =   2^7 3^2 5^2 7^2
       216     2177280  =   2^8 3^5 5 7
       216     3225600  =   2^11 3^2 5^2 7
       216     554400  =   2^5 3^2 5^2 7 11
       216     7776000  =   2^8 3^5 5^3
       216     887040  =   2^8 3^2 5 7 11
       216     900900  =   2^2 3^2 5^2 7 11 13
       216     970200  =   2^3 3^2 5^2 7^2 11
       220     2903040  =   2^10 3^4 5 7
       224     1512000  =   2^6 3^3 5^3 7
       224     3265920  =   2^7 3^6 5 7
       224     665280  =   2^6 3^3 5 7 11
       224     7741440  =   2^13 3^3 5 7
       224     9461760  =   2^13 3 5 7 11
       224     960960  =   2^6 3 5 7 11 13
       225     1587600  =   2^4 3^4 5^2 7^2
       234     6451200  =   2^12 3^2 5^2 7
       240     1774080  =   2^9 3^2 5 7 11
       240     1814400  =   2^7 3^4 5^2 7
       240     2268000  =   2^5 3^4 5^3 7
       240     2419200  =   2^9 3^3 5^2 7
       240     2646000  =   2^4 3^3 5^3 7^2
       240     4354560  =   2^9 3^5 5 7
       240     5806080  =   2^11 3^4 5 7
       240     720720  =   2^4 3^2 5 7 11 13
       240     831600  =   2^4 3^3 5^2 7 11
       240     997920  =   2^5 3^4 5 7 11
       243     2822400  =   2^8 3^2 5^2 7^2
       243     5336100  =   2^2 3^2 5^2 7^2 11^2
       250     5670000  =   2^4 3^4 5^4 7
       252     1108800  =   2^6 3^2 5^2 7 11
       252     2116800  =   2^6 3^3 5^2 7^2
       252     2721600  =   2^6 3^5 5^2 7
       252     6531840  =   2^8 3^6 5 7
       256     1081080  =   2^3 3^3 5 7 11 13
       256     1330560  =   2^7 3^3 5 7 11
       256     1921920  =   2^7 3 5 7 11 13
       256     2042040  =   2^3 3 5 7 11 13 17
       256     2079000  =   2^3 3^3 5^3 7 11
       256     3024000  =   2^7 3^3 5^3 7
       256     9261000  =   2^3 3^3 5^3 7^3
       256     9797760  =   2^7 3^7 5 7
       264     3548160  =   2^10 3^2 5 7 11
       264     4838400  =   2^10 3^3 5^2 7
       264     8709120  =   2^10 3^5 5 7
       270     1940400  =   2^4 3^2 5^2 7^2 11
       270     3175200  =   2^5 3^4 5^2 7^2
       270     3628800  =   2^8 3^4 5^2 7
       270     5644800  =   2^9 3^2 5^2 7^2
       280     1995840  =   2^6 3^4 5 7 11
       280     4536000  =   2^6 3^4 5^3 7
       288     1441440  =   2^5 3^2 5 7 11 13
       288     1663200  =   2^5 3^3 5^2 7 11
       288     1801800  =   2^3 3^2 5^2 7 11 13
       288     2217600  =   2^7 3^2 5^2 7 11
       288     2661120  =   2^8 3^3 5 7 11
       288     2910600  =   2^3 3^3 5^2 7^2 11
       288     2993760  =   2^5 3^5 5 7 11
       288     3063060  =   2^2 3^2 5 7 11 13 17
       288     3843840  =   2^8 3 5 7 11 13
       288     4233600  =   2^7 3^3 5^2 7^2
       288     5292000  =   2^5 3^3 5^3 7^2
       288     5443200  =   2^7 3^5 5^2 7
       288     6048000  =   2^8 3^3 5^3 7
       288     6804000  =   2^5 3^5 5^3 7
       288     7096320  =   2^11 3^2 5 7 11
       288     9676800  =   2^11 3^3 5^2 7
       294     8164800  =   2^6 3^6 5^2 7
       300     2494800  =   2^4 3^4 5^2 7 11
       300     7257600  =   2^9 3^4 5^2 7
       300     7938000  =   2^4 3^4 5^3 7^2
       315     6350400  =   2^6 3^4 5^2 7^2
       320     2162160  =   2^4 3^3 5 7 11 13
       320     3991680  =   2^7 3^4 5 7 11
       320     4084080  =   2^4 3 5 7 11 13 17
       320     4158000  =   2^4 3^3 5^3 7 11
       320     5322240  =   2^9 3^3 5 7 11
       320     7687680  =   2^9 3 5 7 11 13
       320     9072000  =   2^7 3^4 5^3 7
       324     3880800  =   2^5 3^2 5^2 7^2 11
       324     4435200  =   2^8 3^2 5^2 7 11
       324     6306300  =   2^2 3^2 5^2 7^2 11 13
       324     8467200  =   2^8 3^3 5^2 7^2
       324     9525600  =   2^5 3^5 5^2 7^2
       336     2882880  =   2^6 3^2 5 7 11 13
       336     3326400  =   2^6 3^3 5^2 7 11
       336     5987520  =   2^6 3^5 5 7 11
       360     3603600  =   2^4 3^2 5^2 7 11 13
       360     4989600  =   2^5 3^4 5^2 7 11
       360     5821200  =   2^4 3^3 5^2 7^2 11
       360     7983360  =   2^8 3^4 5 7 11
       360     8870400  =   2^9 3^2 5^2 7 11
       378     7761600  =   2^6 3^2 5^2 7^2 11
       384     4324320  =   2^5 3^3 5 7 11 13
       384     5405400  =   2^3 3^3 5^2 7 11 13
       384     5765760  =   2^7 3^2 5 7 11 13
       384     6126120  =   2^3 3^2 5 7 11 13 17
       384     6652800  =   2^7 3^3 5^2 7 11
       384     8168160  =   2^5 3 5 7 11 13 17
       384     8316000  =   2^5 3^3 5^3 7 11
       400     6486480  =   2^4 3^4 5 7 11 13
       420     9979200  =   2^6 3^4 5^2 7 11
       432     7207200  =   2^5 3^2 5^2 7 11 13
       448     8648640  =   2^6 3^3 5 7 11 13
       d(n)    n            factored

====================================

int main()
{
  system("date");

  for(int e2 = 0; e2 <= 23; e2++){
  for(int e3 = 0; e3 <= e2; e3++){
  for(int e5 = 0; e5 <= e3; e5++){
  for(int e7 = 0; e7 <= e5; e7++){
  for(int e11 = 0; e11 <= e7; e11++){
  for(int e13 = 0; e13 <= e11; e13++){
  for(int e17 = 0; e17 <= e13; e17++){

     mpz_class n = to_power(2,e2);
     n *= to_power(3,e3);
         n *= to_power(5,e5);
     n *= to_power(7,e7);
         n *= to_power(11,e11);
           n *= to_power(13,e13);
          n *= to_power(17,e17);
      if( n < 10000000)
      {
         cout << setw(10) <<  mp_Divisor_Count(n) << "     ";
         cout << n << "  =  " << mp_Factored(n) << endl;
      }
    }}}}}}}

  system("date");
  return 0;
}

=======================================