Records and first position of records in A252665
by Michael Thomas De Vlieger, St. Louis, MO 201708281445, revised 201708311730.

k = record-setting value in A252665
m = first position in A252665 of k. 
m followed by "a" or "b" appear in A002182 or A007416 respectively.
A054841(m) = multiplicity notation of m. "211" is read as 2^2 * 3 * 5 = 60.

 n         m               k   A054841(m)
 1         1   a   b       1   0
 2         4   a   b       2   2
 3         8               3   3
 4        12   a   b       4   21
 5        16       b       5   4
 6        24   a   b       7   31
 7        36   a   b       9   22
 8        48   a   b      12   41
 9        72              16   32
10        96              18   51
11       120   a   b      21   311
12       144       b      28   42
13       216              30   33
14       240   a   b      37   411
15       288              43   52
16       360   a   b      51   321
17       432              53   43
18       480              59   511
19       576       b      66   62
20       720   a   b      92   421
21      1080             103   331
22      1440             150   521
23      2160             188   431
24      2520   a   b     189   3211
25      2880       b     235   621
26      3600       b     239   422
27      4320             312   531
28      5040   a   b     351   4211
29      7200             396   522
30      7560   a   b     400   3311
31      8640             493   631
32     10080   a   b     593   5211
33     14400       b     628   622
34     15120   a   b     751   4311
35     20160   a   b     947   6211
36     25200   a   b     954   4221
37     30240            1283   5311
38     40320            1433   7211
39     50400   a   b    1632   5221
40     60480       b    2063   6311
41     75600            2074   4321
42     80640            2088   8211
43     90720            2446   5411
44    100800       b    2629   6221
45    120960            3143   7311
46    151200            3582   5321
47    181440       b    3952   6411
48    201600            4008   7221
49    221760   a   b    4121   62111
50    241920            4602   8311
51    302400            5803   6321
52    362880            6046   7411
53    443520            6323   72111
54    453600            6899   5421
55    554400   a   b    7217   52211
56    604800            8898   7321
57    665280   a   b    9203   63111
58    831600            9235   43211
59    887040            9309   82111
60    907200       b   11220   6421
61   1108800       b   11765   62211
62   1209600           13097   8321
63   1330560           14183   73111
64   1663200           16199   53211
65   1814400           17261   7421
66   1995840       b   17908   64111
67   2217600           18143   72211
68   2419200           18612   9321
69   2661120           20952   83111
70   3326400           26513   63211
71   3991680           27676   74111
72   4989600           31630   54211
73   6350400       b   32249   6422
74   6652800           41033   73211
75   8648640   a   b   42833   631111
76   9979200       b   51904   64211


Remarks and observations:

1. This analysis is based on terms 1 <= m <= 10^7.

2. Though m contains many terms of A002182, {2, 6, 60, 180, 840, 1260, 1680, 27720, 45360, 55440, 
   83160, 110880, 166320, 277200, 332640, 498960, 720720, 1081080, 1441440, 2162160, 2882880, 
   3603600, 4324320, 6486480, 7207200, ...} are not found in m, and m contains terms that are not 
   found in A002182. The trend across A218320 and A252665 is that increasingly many terms
   1 <= m <= 10^7 are not in A002182.
   A few more terms m (i.e., 16, 144, 576, ...) appear in A007416.

Conjectures:

1. Numbers in m are products of the first several consecutive primes.

2. The largest prime factor of m generally has multiplicity 1.

3. The multiplicities of prime factors p of m generally decrease or stay the same as p increases.