# Smallest number with at least $n$ divisors.

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 ?

• Maybe you're looking for Highly Composite Numbers? – Matti P. Apr 26 '18 at 10:37
• OEIS A061799 - which looks rather familiar to me - does not give much. The deduplicated OEIS A002182 has more links and may be more helpful – Henry Apr 26 '18 at 10:50
• Your question is not quite clear. You say that the smallest number that has 11 divisors is 1024 and the smallest number with 12 divisors is 60. Which means that the smallest number with 11 divisors is not 1024 but 60. Are you asking for the smallest number that has exactly N divisors or at least N divisors? – Oldboy Apr 26 '18 at 12:54

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;
}


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

• I appreciate the help, but it seems quite complicated... first you mentioned about primorial numbers... and you say "that would mean 2^k is larger than you want..." that went over my head.... next you mention " prime number theorem say LogL ~ W" ... well what's W? A bit detailed explanation could have been more helpful... thanks for the help though... :) – Aizen Apr 27 '18 at 17:13
• @Aizen I wrote a small program myself, numbers smaller than 10,000,000 with prime factors up to 17 and non-increasing exponents. Take a look at the output. After running, I used a Unix sort -n command to go from least number of divisors to most. If there is more than one way to get a certain number of divisors, the numbers $n$ that work are not necessarily in order, so you need to be aware of that. It is more work to force both d(n) and n to be printed out in correct order, as in a dictionary – Will Jagy Apr 27 '18 at 18:24
• Jaggy How long did your programe take ?? – Aizen Apr 29 '18 at 5:52
• @Aizen The command system("date") prints out the time as on an ordinary human clock. When I sorted the file, both "date" lines went to the start of the file. It took 26 seconds; about half a minute as I watched. – Will Jagy Apr 29 '18 at 13:41