[eslxtlcv] variations on highly composite numbers

highly composite numbers are numbers whose number of divisors exceeds the previous record.  they tend to be very smooth numbers.  here is Pari/GP code from OEIS A002182:

print1(r=1); forstep(n=2, 1e7, 2, if(numdiv(n)>r, r=numdiv(n); print1(", "n)))

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 6486480, 7207200, 8648640

(why is it safe to only consider even numbers?)

loosen the criterion: number of divisors equals or exceeds previous record:

print1(r=1); for(n=2, 1e7, if(numdiv(n)>=r, r=numdiv(n); print1(", "n)))

1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720, 30240, 32760, 36960, 37800, 40320, 41580, 42840, 43680, 45360, 50400, 55440, 65520, 75600, 83160, 98280, 110880, 131040, 138600, 151200, 163800, 166320, 196560, 221760, 262080, 277200, 327600, 332640, 360360, 393120, 415800, 443520, 471240, 480480, 491400, 498960, 554400, 655200, 665280, 720720, 831600, 942480, 982800, 997920, 1053360, 1081080, 1330560, 1413720, 1441440, 1663200, 1801800, 1884960, 1965600, 2106720, 2162160, 2827440, 2882880, 3326400, 3603600, 4324320, 5405400, 5654880, 5765760, 6126120, 6320160, 6486480, 7207200, 8648640

(are 1 and 3 the only odd numbers in the above list?)

(the ElevenSmooth factoring project targeted Mersenne number 3326400.)

further loosen the criterion: number of divisors sets a new record for second place, or ties first place.  second place number of divisors is a number strictly less than first place.

print1(1); r1=1; r2=0; for(n=2, 1e7, nd=numdiv(n); if(nd>r2, print1(", "n); if(nd>r1, r2=r1; r1=nd, if(r2<nd && nd<r1, r2=nd))))

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 240, 336, 360, 420, 480, 504, 540, 576, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 2880, 3360, 3600, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 6300, 6720, 7560, 9240, 10080, 12600, 13860, 15120, 18480, 20160, 25200, 27720, 30240, 32760, 36960, 37800, 40320, 41580, 42840, 43680, 45360, 50400, 55440, 60480, 65520, 75600, 83160, 98280, 100800, 110880, 131040, 138600, 151200, 163800, 166320, 196560, 221760, 262080, 277200, 327600, 332640, 360360, 393120, 415800, 443520, 471240, 480480, 491400, 498960, 554400, 655200, 665280, 720720, 831600, 942480, 982800, 997920, 1053360, 1081080, 1108800, 1330560, 1413720, 1441440, 1663200, 1801800, 1884960, 1940400, 1965600, 1995840, 2106720, 2162160, 2494800, 2827440, 2882880, 3326400, 3603600, 4324320, 5405400, 5654880, 5765760, 6126120, 6320160, 6486480, 7207200, 8648640

further loosen the criterion: number of divisors equals or exceeds second place.  we give each number, its number of divisors, and prime factorization.  we continue the list until 29 appears as a factor.

print(1" "numdiv(1)" "factorint(1)); r1=1; r2=0; for(n=2, +oo, nd=numdiv(n); if(nd>=r2, f=factorint(n); print(n" "nd" "f); sf=matsize(f)[1]; if(f[sf,1]>23, break); if(nd>r1, r2=r1; r1=nd, if(r2<nd && nd<r1, r2=nd))))

1 1 matrix(0,2)
2 2 Mat([2, 1])
3 2 Mat([3, 1])
4 3 Mat([2, 2])
5 2 Mat([5, 1])
6 4 [2, 1; 3, 1]
8 4 Mat([2, 3])
9 3 Mat([3, 2])
10 4 [2, 1; 5, 1]
12 6 [2, 2; 3, 1]
14 4 [2, 1; 7, 1]
15 4 [3, 1; 5, 1]
16 5 Mat([2, 4])
18 6 [2, 1; 3, 2]
20 6 [2, 2; 5, 1]
24 8 [2, 3; 3, 1]
28 6 [2, 2; 7, 1]
30 8 [2, 1; 3, 1; 5, 1]
32 6 Mat([2, 5])
36 9 [2, 2; 3, 2]
40 8 [2, 3; 5, 1]
42 8 [2, 1; 3, 1; 7, 1]
48 10 [2, 4; 3, 1]
60 12 [2, 2; 3, 1; 5, 1]
72 12 [2, 3; 3, 2]
80 10 [2, 4; 5, 1]
84 12 [2, 2; 3, 1; 7, 1]
90 12 [2, 1; 3, 2; 5, 1]
96 12 [2, 5; 3, 1]
108 12 [2, 2; 3, 3]
112 10 [2, 4; 7, 1]
120 16 [2, 3; 3, 1; 5, 1]
126 12 [2, 1; 3, 2; 7, 1]
132 12 [2, 2; 3, 1; 11, 1]
140 12 [2, 2; 5, 1; 7, 1]
144 15 [2, 4; 3, 2]
168 16 [2, 3; 3, 1; 7, 1]
180 18 [2, 2; 3, 2; 5, 1]
210 16 [2, 1; 3, 1; 5, 1; 7, 1]
216 16 [2, 3; 3, 3]
240 20 [2, 4; 3, 1; 5, 1]
252 18 [2, 2; 3, 2; 7, 1]
288 18 [2, 5; 3, 2]
300 18 [2, 2; 3, 1; 5, 2]
336 20 [2, 4; 3, 1; 7, 1]
360 24 [2, 3; 3, 2; 5, 1]
420 24 [2, 2; 3, 1; 5, 1; 7, 1]
432 20 [2, 4; 3, 3]
480 24 [2, 5; 3, 1; 5, 1]
504 24 [2, 3; 3, 2; 7, 1]
528 20 [2, 4; 3, 1; 11, 1]
540 24 [2, 2; 3, 3; 5, 1]
560 20 [2, 4; 5, 1; 7, 1]
576 21 [2, 6; 3, 2]
600 24 [2, 3; 3, 1; 5, 2]
630 24 [2, 1; 3, 2; 5, 1; 7, 1]
660 24 [2, 2; 3, 1; 5, 1; 11, 1]
672 24 [2, 5; 3, 1; 7, 1]
720 30 [2, 4; 3, 2; 5, 1]
756 24 [2, 2; 3, 3; 7, 1]
780 24 [2, 2; 3, 1; 5, 1; 13, 1]
792 24 [2, 3; 3, 2; 11, 1]
840 32 [2, 3; 3, 1; 5, 1; 7, 1]
1008 30 [2, 4; 3, 2; 7, 1]
1080 32 [2, 3; 3, 3; 5, 1]
1200 30 [2, 4; 3, 1; 5, 2]
1260 36 [2, 2; 3, 2; 5, 1; 7, 1]
1320 32 [2, 3; 3, 1; 5, 1; 11, 1]
1440 36 [2, 5; 3, 2; 5, 1]
1512 32 [2, 3; 3, 3; 7, 1]
1560 32 [2, 3; 3, 1; 5, 1; 13, 1]
1680 40 [2, 4; 3, 1; 5, 1; 7, 1]
1800 36 [2, 3; 3, 2; 5, 2]
1980 36 [2, 2; 3, 2; 5, 1; 11, 1]
2016 36 [2, 5; 3, 2; 7, 1]
2100 36 [2, 2; 3, 1; 5, 2; 7, 1]
2160 40 [2, 4; 3, 3; 5, 1]
2340 36 [2, 2; 3, 2; 5, 1; 13, 1]
2400 36 [2, 5; 3, 1; 5, 2]
2520 48 [2, 3; 3, 2; 5, 1; 7, 1]
2640 40 [2, 4; 3, 1; 5, 1; 11, 1]
2880 42 [2, 6; 3, 2; 5, 1]
3360 48 [2, 5; 3, 1; 5, 1; 7, 1]
3600 45 [2, 4; 3, 2; 5, 2]
3780 48 [2, 2; 3, 3; 5, 1; 7, 1]
3960 48 [2, 3; 3, 2; 5, 1; 11, 1]
4200 48 [2, 3; 3, 1; 5, 2; 7, 1]
4320 48 [2, 5; 3, 3; 5, 1]
4620 48 [2, 2; 3, 1; 5, 1; 7, 1; 11, 1]
4680 48 [2, 3; 3, 2; 5, 1; 13, 1]
5040 60 [2, 4; 3, 2; 5, 1; 7, 1]
5280 48 [2, 5; 3, 1; 5, 1; 11, 1]
5400 48 [2, 3; 3, 3; 5, 2]
5460 48 [2, 2; 3, 1; 5, 1; 7, 1; 13, 1]
5544 48 [2, 3; 3, 2; 7, 1; 11, 1]
5760 48 [2, 7; 3, 2; 5, 1]
5880 48 [2, 3; 3, 1; 5, 1; 7, 2]
5940 48 [2, 2; 3, 3; 5, 1; 11, 1]
6048 48 [2, 5; 3, 3; 7, 1]
6120 48 [2, 3; 3, 2; 5, 1; 17, 1]
6240 48 [2, 5; 3, 1; 5, 1; 13, 1]
6300 54 [2, 2; 3, 2; 5, 2; 7, 1]
6720 56 [2, 6; 3, 1; 5, 1; 7, 1]
7560 64 [2, 3; 3, 3; 5, 1; 7, 1]
7920 60 [2, 4; 3, 2; 5, 1; 11, 1]
8400 60 [2, 4; 3, 1; 5, 2; 7, 1]
9240 64 [2, 3; 3, 1; 5, 1; 7, 1; 11, 1]
9360 60 [2, 4; 3, 2; 5, 1; 13, 1]
10080 72 [2, 5; 3, 2; 5, 1; 7, 1]
10920 64 [2, 3; 3, 1; 5, 1; 7, 1; 13, 1]
11880 64 [2, 3; 3, 3; 5, 1; 11, 1]
12600 72 [2, 3; 3, 2; 5, 2; 7, 1]
13440 64 [2, 7; 3, 1; 5, 1; 7, 1]
13860 72 [2, 2; 3, 2; 5, 1; 7, 1; 11, 1]
14040 64 [2, 3; 3, 3; 5, 1; 13, 1]
14280 64 [2, 3; 3, 1; 5, 1; 7, 1; 17, 1]
15120 80 [2, 4; 3, 3; 5, 1; 7, 1]
15840 72 [2, 5; 3, 2; 5, 1; 11, 1]
16380 72 [2, 2; 3, 2; 5, 1; 7, 1; 13, 1]
16800 72 [2, 5; 3, 1; 5, 2; 7, 1]
17640 72 [2, 3; 3, 2; 5, 1; 7, 2]
18480 80 [2, 4; 3, 1; 5, 1; 7, 1; 11, 1]
18720 72 [2, 5; 3, 2; 5, 1; 13, 1]
18900 72 [2, 2; 3, 3; 5, 2; 7, 1]
19800 72 [2, 3; 3, 2; 5, 2; 11, 1]
20160 84 [2, 6; 3, 2; 5, 1; 7, 1]
21840 80 [2, 4; 3, 1; 5, 1; 7, 1; 13, 1]
22680 80 [2, 3; 3, 4; 5, 1; 7, 1]
23760 80 [2, 4; 3, 3; 5, 1; 11, 1]
25200 90 [2, 4; 3, 2; 5, 2; 7, 1]
27720 96 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1]
30240 96 [2, 5; 3, 3; 5, 1; 7, 1]
32760 96 [2, 3; 3, 2; 5, 1; 7, 1; 13, 1]
35280 90 [2, 4; 3, 2; 5, 1; 7, 2]
36960 96 [2, 5; 3, 1; 5, 1; 7, 1; 11, 1]
37800 96 [2, 3; 3, 3; 5, 2; 7, 1]
39600 90 [2, 4; 3, 2; 5, 2; 11, 1]
40320 96 [2, 7; 3, 2; 5, 1; 7, 1]
41580 96 [2, 2; 3, 3; 5, 1; 7, 1; 11, 1]
42840 96 [2, 3; 3, 2; 5, 1; 7, 1; 17, 1]
43680 96 [2, 5; 3, 1; 5, 1; 7, 1; 13, 1]
45360 100 [2, 4; 3, 4; 5, 1; 7, 1]
46200 96 [2, 3; 3, 1; 5, 2; 7, 1; 11, 1]
47520 96 [2, 5; 3, 3; 5, 1; 11, 1]
47880 96 [2, 3; 3, 2; 5, 1; 7, 1; 19, 1]
49140 96 [2, 2; 3, 3; 5, 1; 7, 1; 13, 1]
50400 108 [2, 5; 3, 2; 5, 2; 7, 1]
55440 120 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1]
60480 112 [2, 6; 3, 3; 5, 1; 7, 1]
65520 120 [2, 4; 3, 2; 5, 1; 7, 1; 13, 1]
73920 112 [2, 6; 3, 1; 5, 1; 7, 1; 11, 1]
75600 120 [2, 4; 3, 3; 5, 2; 7, 1]
83160 128 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1]
85680 120 [2, 4; 3, 2; 5, 1; 7, 1; 17, 1]
90720 120 [2, 5; 3, 4; 5, 1; 7, 1]
92400 120 [2, 4; 3, 1; 5, 2; 7, 1; 11, 1]
95760 120 [2, 4; 3, 2; 5, 1; 7, 1; 19, 1]
98280 128 [2, 3; 3, 3; 5, 1; 7, 1; 13, 1]
100800 126 [2, 6; 3, 2; 5, 2; 7, 1]
110880 144 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1]
120120 128 [2, 3; 3, 1; 5, 1; 7, 1; 11, 1; 13, 1]
120960 128 [2, 7; 3, 3; 5, 1; 7, 1]
128520 128 [2, 3; 3, 3; 5, 1; 7, 1; 17, 1]
131040 144 [2, 5; 3, 2; 5, 1; 7, 1; 13, 1]
138600 144 [2, 3; 3, 2; 5, 2; 7, 1; 11, 1]
143640 128 [2, 3; 3, 3; 5, 1; 7, 1; 19, 1]
147840 128 [2, 7; 3, 1; 5, 1; 7, 1; 11, 1]
151200 144 [2, 5; 3, 3; 5, 2; 7, 1]
154440 128 [2, 3; 3, 3; 5, 1; 11, 1; 13, 1]
157080 128 [2, 3; 3, 1; 5, 1; 7, 1; 11, 1; 17, 1]
163800 144 [2, 3; 3, 2; 5, 2; 7, 1; 13, 1]
166320 160 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1]
171360 144 [2, 5; 3, 2; 5, 1; 7, 1; 17, 1]
180180 144 [2, 2; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
184800 144 [2, 5; 3, 1; 5, 2; 7, 1; 11, 1]
191520 144 [2, 5; 3, 2; 5, 1; 7, 1; 19, 1]
194040 144 [2, 3; 3, 2; 5, 1; 7, 2; 11, 1]
196560 160 [2, 4; 3, 3; 5, 1; 7, 1; 13, 1]
201600 144 [2, 7; 3, 2; 5, 2; 7, 1]
205920 144 [2, 5; 3, 2; 5, 1; 11, 1; 13, 1]
207900 144 [2, 2; 3, 3; 5, 2; 7, 1; 11, 1]
211680 144 [2, 5; 3, 3; 5, 1; 7, 2]
214200 144 [2, 3; 3, 2; 5, 2; 7, 1; 17, 1]
218400 144 [2, 5; 3, 1; 5, 2; 7, 1; 13, 1]
221760 168 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1]
240240 160 [2, 4; 3, 1; 5, 1; 7, 1; 11, 1; 13, 1]
249480 160 [2, 3; 3, 4; 5, 1; 7, 1; 11, 1]
257040 160 [2, 4; 3, 3; 5, 1; 7, 1; 17, 1]
262080 168 [2, 6; 3, 2; 5, 1; 7, 1; 13, 1]
277200 180 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1]
302400 168 [2, 6; 3, 3; 5, 2; 7, 1]
327600 180 [2, 4; 3, 2; 5, 2; 7, 1; 13, 1]
332640 192 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1]
360360 192 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
388080 180 [2, 4; 3, 2; 5, 1; 7, 2; 11, 1]
393120 192 [2, 5; 3, 3; 5, 1; 7, 1; 13, 1]
415800 192 [2, 3; 3, 3; 5, 2; 7, 1; 11, 1]
428400 180 [2, 4; 3, 2; 5, 2; 7, 1; 17, 1]
443520 192 [2, 7; 3, 2; 5, 1; 7, 1; 11, 1]
453600 180 [2, 5; 3, 4; 5, 2; 7, 1]
458640 180 [2, 4; 3, 2; 5, 1; 7, 2; 13, 1]
471240 192 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 17, 1]
478800 180 [2, 4; 3, 2; 5, 2; 7, 1; 19, 1]
480480 192 [2, 5; 3, 1; 5, 1; 7, 1; 11, 1; 13, 1]
491400 192 [2, 3; 3, 3; 5, 2; 7, 1; 13, 1]
498960 200 [2, 4; 3, 4; 5, 1; 7, 1; 11, 1]
514080 192 [2, 5; 3, 3; 5, 1; 7, 1; 17, 1]
524160 192 [2, 7; 3, 2; 5, 1; 7, 1; 13, 1]
526680 192 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 19, 1]
540540 192 [2, 2; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
554400 216 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1]
589680 200 [2, 4; 3, 4; 5, 1; 7, 1; 13, 1]
655200 216 [2, 5; 3, 2; 5, 2; 7, 1; 13, 1]
665280 224 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1]
720720 240 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
786240 224 [2, 6; 3, 3; 5, 1; 7, 1; 13, 1]
831600 240 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1]
942480 240 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 17, 1]
960960 224 [2, 6; 3, 1; 5, 1; 7, 1; 11, 1; 13, 1]
982800 240 [2, 4; 3, 3; 5, 2; 7, 1; 13, 1]
997920 240 [2, 5; 3, 4; 5, 1; 7, 1; 11, 1]
1028160 224 [2, 6; 3, 3; 5, 1; 7, 1; 17, 1]
1053360 240 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 19, 1]
1081080 256 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
1108800 252 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1]
1310400 252 [2, 6; 3, 2; 5, 2; 7, 1; 13, 1]
1330560 256 [2, 7; 3, 3; 5, 1; 7, 1; 11, 1]
1413720 256 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1]
1441440 288 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
1572480 256 [2, 7; 3, 3; 5, 1; 7, 1; 13, 1]
1580040 256 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 19, 1]
1663200 288 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1]
1670760 256 [2, 3; 3, 3; 5, 1; 7, 1; 13, 1; 17, 1]
1801800 288 [2, 3; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1]
1867320 256 [2, 3; 3, 3; 5, 1; 7, 1; 13, 1; 19, 1]
1884960 288 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 17, 1]
1912680 256 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 23, 1]
1921920 256 [2, 7; 3, 1; 5, 1; 7, 1; 11, 1; 13, 1]
1940400 270 [2, 4; 3, 2; 5, 2; 7, 2; 11, 1]
1965600 288 [2, 5; 3, 3; 5, 2; 7, 1; 13, 1]
1995840 280 [2, 6; 3, 4; 5, 1; 7, 1; 11, 1]
2106720 288 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 19, 1]
2162160 320 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
2217600 288 [2, 7; 3, 2; 5, 2; 7, 1; 11, 1]
2227680 288 [2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 17, 1]
2328480 288 [2, 5; 3, 3; 5, 1; 7, 2; 11, 1]
2356200 288 [2, 3; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1]
2402400 288 [2, 5; 3, 1; 5, 2; 7, 1; 11, 1; 13, 1]
2489760 288 [2, 5; 3, 2; 5, 1; 7, 1; 13, 1; 19, 1]
2494800 300 [2, 4; 3, 4; 5, 2; 7, 1; 11, 1]
2827440 320 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1]
2882880 336 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
3160080 320 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1; 19, 1]
3243240 320 [2, 3; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1]
3326400 336 [2, 6; 3, 3; 5, 2; 7, 1; 11, 1]
3341520 320 [2, 4; 3, 3; 5, 1; 7, 1; 13, 1; 17, 1]
3603600 360 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1]
3769920 336 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 17, 1]
3931200 336 [2, 6; 3, 3; 5, 2; 7, 1; 13, 1]
4213440 336 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 19, 1]
4324320 384 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
4712400 360 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1]
4989600 360 [2, 5; 3, 4; 5, 2; 7, 1; 11, 1]
5045040 360 [2, 4; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1]
5266800 360 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 19, 1]
5405400 384 [2, 3; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1]
5569200 360 [2, 4; 3, 2; 5, 2; 7, 1; 13, 1; 17, 1]
5654880 384 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1]
5765760 384 [2, 7; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1]
5821200 360 [2, 4; 3, 3; 5, 2; 7, 2; 11, 1]
5896800 360 [2, 5; 3, 4; 5, 2; 7, 1; 13, 1]
6126120 384 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
6224400 360 [2, 4; 3, 2; 5, 2; 7, 1; 13, 1; 19, 1]
6320160 384 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 19, 1]
6375600 360 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 23, 1]
6486480 400 [2, 4; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1]
6652800 384 [2, 7; 3, 3; 5, 2; 7, 1; 11, 1]
6683040 384 [2, 5; 3, 3; 5, 1; 7, 1; 13, 1; 17, 1]
6846840 384 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
7068600 384 [2, 3; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1]
7207200 432 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1]
8482320 400 [2, 4; 3, 4; 5, 1; 7, 1; 11, 1; 17, 1]
8648640 448 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
9424800 432 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1]
10090080 432 [2, 5; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1]
10533600 432 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 19, 1]
10810800 480 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1]
11309760 448 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1]
12252240 480 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
12640320 448 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 19, 1]
12972960 480 [2, 5; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1]
13366080 448 [2, 6; 3, 3; 5, 1; 7, 1; 13, 1; 17, 1]
13693680 480 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
14137200 480 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1]
14414400 504 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1]
15135120 480 [2, 4; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1]
15800400 480 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 19, 1]
16216200 480 [2, 3; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1]
16576560 480 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
16707600 480 [2, 4; 3, 3; 5, 2; 7, 1; 13, 1; 17, 1]
16964640 480 [2, 5; 3, 4; 5, 1; 7, 1; 11, 1; 17, 1]
17297280 512 [2, 7; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
18378360 512 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
18849600 504 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1]
20180160 504 [2, 6; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1]
20540520 512 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
21067200 504 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 19, 1]
21621600 576 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1]
22619520 512 [2, 7; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1]
24504480 576 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
24864840 512 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
25225200 540 [2, 4; 3, 2; 5, 2; 7, 2; 11, 1; 13, 1]
25945920 560 [2, 6; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1]
27387360 576 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
28274400 576 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1]
28828800 576 [2, 7; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1]
30270240 576 [2, 5; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1]
30630600 576 [2, 3; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
31600800 576 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 19, 1]
32432400 600 [2, 4; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1]
33153120 576 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
33415200 576 [2, 5; 3, 3; 5, 2; 7, 1; 13, 1; 17, 1]
34234200 576 [2, 3; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
34594560 576 [2, 8; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1]
35814240 576 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 17, 1; 19, 1]
36036000 576 [2, 5; 3, 2; 5, 3; 7, 1; 11, 1; 13, 1]
36756720 640 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
41081040 640 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
42411600 600 [2, 4; 3, 4; 5, 2; 7, 1; 11, 1; 17, 1]
43243200 672 [2, 6; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1]
49008960 672 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
49729680 640 [2, 4; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
50450400 648 [2, 5; 3, 2; 5, 2; 7, 2; 11, 1; 13, 1]
54774720 672 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
56548800 672 [2, 6; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1]
57657600 648 [2, 8; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1]
60540480 672 [2, 6; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1]
61261200 720 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
63201600 672 [2, 6; 3, 3; 5, 2; 7, 1; 11, 1; 19, 1]
64864800 720 [2, 5; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1]
66306240 672 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
66830400 672 [2, 6; 3, 3; 5, 2; 7, 1; 13, 1; 17, 1]
68468400 720 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
71628480 672 [2, 6; 3, 2; 5, 1; 7, 1; 11, 1; 17, 1; 19, 1]
72072000 672 [2, 6; 3, 2; 5, 3; 7, 1; 11, 1; 13, 1]
73513440 768 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
75675600 720 [2, 4; 3, 3; 5, 2; 7, 2; 11, 1; 13, 1]
82162080 768 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
82882800 720 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 23, 1]
84823200 720 [2, 5; 3, 4; 5, 2; 7, 1; 11, 1; 17, 1]
85765680 720 [2, 4; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1; 17, 1]
86486400 768 [2, 7; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1]
89535600 720 [2, 4; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1; 19, 1]
90810720 720 [2, 5; 3, 4; 5, 1; 7, 2; 11, 1; 13, 1]
91891800 768 [2, 3; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
94802400 720 [2, 5; 3, 4; 5, 2; 7, 1; 11, 1; 19, 1]
95855760 720 [2, 4; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1; 19, 1]
97297200 720 [2, 4; 3, 5; 5, 2; 7, 1; 11, 1; 13, 1]
98017920 768 [2, 7; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
98960400 720 [2, 4; 3, 3; 5, 2; 7, 2; 11, 1; 17, 1]
99459360 768 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
100245600 720 [2, 5; 3, 4; 5, 2; 7, 1; 13, 1; 17, 1]
100900800 756 [2, 6; 3, 2; 5, 2; 7, 2; 11, 1; 13, 1]
102702600 768 [2, 3; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
107442720 768 [2, 5; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1; 19, 1]
108108000 768 [2, 5; 3, 3; 5, 3; 7, 1; 11, 1; 13, 1]
109549440 768 [2, 7; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
110270160 800 [2, 4; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
113097600 768 [2, 7; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1]
116396280 768 [2, 3; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
121080960 768 [2, 7; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1]
122522400 864 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
123243120 800 [2, 4; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
129729600 840 [2, 6; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1]
136936800 864 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
147026880 896 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
151351200 864 [2, 5; 3, 3; 5, 2; 7, 2; 11, 1; 13, 1]
164324160 896 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
165765600 864 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 23, 1]
171531360 864 [2, 5; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1; 17, 1]
172972800 864 [2, 8; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1]
179071200 864 [2, 5; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1; 19, 1]
183783600 960 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
198918720 896 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
205405200 960 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
214885440 896 [2, 6; 3, 3; 5, 1; 7, 1; 11, 1; 17, 1; 19, 1]
216216000 896 [2, 6; 3, 3; 5, 3; 7, 1; 11, 1; 13, 1]
220540320 960 [2, 5; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
227026800 900 [2, 4; 3, 4; 5, 2; 7, 2; 11, 1; 13, 1]
232792560 960 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
245044800 1008 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
246486240 960 [2, 5; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
248648400 960 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 23, 1]
257297040 960 [2, 4; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1; 17, 1]
259459200 960 [2, 7; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1]
268606800 960 [2, 4; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1; 19, 1]
273873600 1008 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
275675400 960 [2, 3; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
281801520 960 [2, 4; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 23, 1]
287567280 960 [2, 4; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1; 19, 1]
294053760 1024 [2, 7; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
302702400 1008 [2, 6; 3, 3; 5, 2; 7, 2; 11, 1; 13, 1]
328648320 1024 [2, 7; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
331531200 1008 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 23, 1]
343062720 1008 [2, 6; 3, 2; 5, 1; 7, 2; 11, 1; 13, 1; 17, 1]
349188840 1024 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
358142400 1008 [2, 6; 3, 2; 5, 2; 7, 1; 11, 1; 17, 1; 19, 1]
367567200 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
397837440 1024 [2, 7; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 23, 1]
410810400 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
422702280 1024 [2, 3; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 23, 1]
428828400 1080 [2, 4; 3, 2; 5, 2; 7, 2; 11, 1; 13, 1; 17, 1]
441080640 1120 [2, 6; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
465585120 1152 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
490089600 1152 [2, 7; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
492972480 1120 [2, 6; 3, 4; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
497296800 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 23, 1]
514594080 1152 [2, 5; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1; 17, 1]
537213600 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1; 19, 1]
547747200 1152 [2, 7; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
551350800 1200 [2, 4; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1]
563603040 1152 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1; 23, 1]
575134560 1152 [2, 5; 3, 3; 5, 1; 7, 2; 11, 1; 13, 1; 19, 1]
581981400 1152 [2, 3; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1]
588107520 1152 [2, 8; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
605404800 1152 [2, 7; 3, 3; 5, 2; 7, 2; 11, 1; 13, 1]
612612000 1152 [2, 5; 3, 2; 5, 3; 7, 1; 11, 1; 13, 1; 17, 1]
616215600 1200 [2, 4; 3, 4; 5, 2; 7, 1; 11, 1; 13, 1; 19, 1]
627026400 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 29, 1]
cpu time = 18min, 31,610 ms, real time = 18min, 31,802 ms.

the first occurrence of 31 as a factor is only a little bit larger:

627026400 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 29, 1]
629909280 1152 [2, 5; 3, 2; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1; 23, 1]
634888800 1152 [2, 5; 3, 3; 5, 2; 7, 1; 13, 1; 17, 1; 19, 1]
643242600 1152 [2, 3; 3, 3; 5, 2; 7, 2; 11, 1; 13, 1; 17, 1]
650311200 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 17, 1; 23, 1]
657296640 1152 [2, 8; 3, 3; 5, 1; 7, 1; 11, 1; 13, 1; 19, 1]
661620960 1152 [2, 5; 3, 5; 5, 1; 7, 1; 11, 1; 13, 1; 17, 1]
663062400 1152 [2, 7; 3, 2; 5, 2; 7, 1; 11, 1; 13, 1; 23, 1]
670269600 1152 [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 31, 1]

some additional numbers discovered while searching for the first factor of 37 (which we did not find):

670269600 684684000 686125440 696215520 698377680 735134400 821620800 845404560 857656800 931170240 958557600 980179200 994593600 1029188160 1074427200 1095494400 1102701600 1127206080 1150269120 1163962800 1225224000 1232431200 1254052800 1259818560 1269777600 1286485200 1300622400 1323241920 1340539200 1369368000 1392431040 1396755360 1409007600 1437836400 1470268800 1491890400 1543782240 1574773200 1611640800 1629547920 1643241600 1654052400 1690809120 1715313600 1745944200 1837836000 1862340480 1889727840 1917115200 1989187200 2054052000 2058376320 2095133040 2113511400 2131889760 2148854400 2205403200 2327925600 2464862400 2572970400 2793510720 2818015200 2875672800 2940537600 3149546400 3259095840 3286483200 3308104800 3381618240 3430627200 3481077600 3491888400 3675672000 3779455680 3859455600 4190266080 4227022800 4313509200 4410806400 4655851200 4724319600 4888643760 4929724800 5072427360 5145940800 5237832600 5329724400 5354228880 5513508000 5587021440 5636030400 5751345600 6299092800 6518191680 6616209600 6763236480 6962155200 6983776800 7351344000 7558911360 7718911200 8147739600 8380532160 8454045600 9311702400 9448639200 9777287520 10144854720 10291881600 10475665200 10659448800 10708457760 11174042880 11272060800 11394583200 11502691200 11639628000 11835663840 11913501600 12221609400 12355912800 12570798240 12598185600 12681068400 12735122400 12864852000 13036383360 13228094880 13232419200 13385572200 13501968480 13526472960 13599986400 13924310400 13967553600 14172958800 14665931280 15437822400

there are a lot of ways this computation could be optimized.

update: the sequence continues

15437822400 16062686640 16295479200 16908091200 18623404800 18897278400 19554575040 19726106400 20583763200 20951330400 21318897600 21416915520 22789166400 23279256000 23671327680 23827003200 24443218800 24711825600 25141596480 25362136800 25470244800 25729704000 26456189760 26771144400 27003936960 27199972800 27935107200 28345917600 29331862560 29589159600 30875644800 31426995600 31978346400 32125373280 32590958400 33816182400 34918884000 37794556800 39109150080 39452212800 40156716600 40505905440 41902660800 42270228000 42637795200 42833831040 43299416160 45578332800 46558512000 47243196000 47342655360 47654006400 48188059920 48886437600 50724273600 53542288800 55870214400 56691835200 58663725120 59178319200 62853991200 63956692800 64250746560 65181916800 66140474400 67509842400 67632364800 69837768000 72165693600 73329656400 80313433200 83805321600 88767478800 96376119840 97772875200 101264763600 101448547200 104756652000 107084577600 108248540400 111740428800 112438806480 113383670400 117327450240 118356638400 120470149800 121517716320 122216094000 122583661200 125707982400 126810684000 127913385600 128501493120 132280948800 135019684800 139675536000 144331387200 146659312800 160626866400 167610643200 177534957600 187398010800 192752239680 195545750400 202529527200 209513304000 214169155200 216497080800 224877612960 236713276800 240940299600 244432188000 245167322400 251415964800 257002986240 258399741600 262075413600 264561897600 267711444000 270039369600 274010536800 279351072000 281097016200 283541338080 286334848800 288662774400 289128359520 292907815200 293318625600 303794290800 321253732800 355069915200 374796021600 391091500800 405059054400 428338310400 432994161600 439977938400 449755225920 472568896800 473426553600 481880599200 488864376000 490334644800 516799483200 524150827200 535422888000 548021073600 562194032400 567082676160 572669697600 578256719040 585815630400 586637251200 591783192000 600604804800 606191826240 607588581600 621090550080 625599374400 628539912000 642507465600 649491242400 674632838880 708853345200 710139830400 722820898800 733296564000 735501967200 749592043200 803134332000 810118108800 865988323200 879955876800 899510451840 931635825120 945137793600 963761198400 977728752000 980669289600 995886571680 1012647636000 1033598966400 1048301654400 1070845776000 1082485404000 1096042147200 1124388064800 1215177163200 1285014931200 1298982484800 1349265677760 1417706690400 1445641797600 1466593128000 1471003934400 1499184086400 1515479565600 1550398449600 1552726375200 1572452481600 1606268664000 1620236217600 1659810952800 1686582097200 1863271650240 1927522396800 2126560035600 2248776129600 2329089562800 2409402996000 2430354326400 2489716429200 2570029862400 2597964969600 2698531355520 2794907475360 2810970162000 2835413380800 2891283595200 2933186256000 2942007868800 2971597028400 2987659715040 2998368172800 3030959131200 3105452750400 3212537328000 3319621905600 3373164194400 3726543300480 3855044793600 3983546286720 4050590544000 4253120071200 4329941616000 4497552259200 4546438696800 4658179125600 4818805992000 4979432858400 5589814950720 5621940324000 5670826761600 5782567190400 5943194056800 5975319430080 6061918262400 6075885816000 6210905500800 6278415343200 6425074656000 6494912424000 6521450775840 6585701522400 6639243811200 6746328388800 6864685027200 6906955255200 6971206001760 6987268688400 7469149287600 7710089587200 8432910486000 8506240142400 8914791085200 8995104518400 9092877393600 9316358251200 9958865716800 10119492583200 10869084626400 11243880648000 11341653523200 11565134380800 11618676669600 11886388113600 12123836524800 12368268712800 12421811001600 12556830686400 12759360213600 13042901551680 13171403044800 13278487622400 13492656777600 13813910510400 13942412003520 13974537376800 14177066904000 14456417976000 14938298575200 14987185012800 15098925441600 15154795656000 15200373988800 15527263752000 15741432907200 16020783979200 16303626939600 16598109528000 16607421230400 16640943359040 16769444852160 16865820972000 16988653281600 17012480284800 17026447838400 17161712568000 17347701571200 17417539339200 17428015004400 17579562960960 17668955304000 17752760625600 17829582170400 17925958290240 17990209036800 18087981912000 18142381857600 18185754787200 18227657448000 18345278952000 18439964262720 18552403069200 18632716502400 18835246029600 19275223968000 19564352327520 19757104567200 19917731433600 20238985166400 21738169252800 22487761296000 23237353339200 23290895628000 23772776227200 24736537425600 24897164292000 25113661372800 25518720427200 26085803103360 26342806089600 26985313555200 27627821020800 27884824007040 27949074753600 29876597150400 32607253879200 33731641944000 34856030008800 35659164340800 37265433004800 37670492059200 39128704655040 39514209134400 39835462867200 40477970332800 41441731531200 41602358397600 41827236010560 41923612130400 42531200712000 43476338505600 43948907402400 44814895725600 44961555038400 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861763138236000 866421317361600 978217616376000 998456601542400 1010824870255200 1034115765883200 1045680900264000 1105434094564800 1145912064897600 1155228423148800 1173861139651200 1201810214404800 1206468393530400 1212989844306240 1224940483166400 1248070751928000 1254817080316800 1284693677467200 1289673110325600 1299631976042400 1304290155168000 1313606513419200 1318467222072000 1336897409047200 1347766493673600 1369504662926400 1378821021177600 1382997319704000 1393808206190400 1394241200352000 1402111916805600 1404200066068800 1429097230360800 1434719170684800 1444035528936000 1467326424564000 1473912126086400 1497684902313600 1498809290378400 1516237305382800

13599986400 is the first occurrence of 37 as a factor (so we had found it in the previous search, but erroneously did not notice its factor of 37).  it has 2304 divisors, and its prime factorization is [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1; 37, 1]

286334848800 is the first occurrence of 41 as a factor.  it has 4608 divisors, and its prime factorization is [2, 5; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1; 41, 1]

600604804800 is the first occurrence of 43 as a factor.  it has 5376 divisors, and its prime factorization is [2, 6; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1; 43, 1]

15098925441600 is the first occurrence of 47 as a factor.  it has 10752 divisors, and its prime factorization is [2, 6; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1; 23, 1; 47, 1]

17026447838400 is the first occurrence of 53 as a factor.  it also has 10752 divisors, and its prime factorization is [2, 6; 3, 3; 5, 2; 7, 1; 11, 1; 13, 1; 17, 1; 19, 1; 23, 1; 53, 1]

we were unable to find a number in this sequence with 59 as a factor.

future post: source code