Records and first position of records in A226378(n), with 0 <= n <= 10^6.
by Michael Thomas De Vlieger, St. Louis, MO 201708242030, revised 201709021100.

k = record-setting value in A226378.
m = first position in A226378 of k. 
m followed by "a" appear in A002182.
m followed by "b" appear in A007416
MN(m) = rev(A054841(m)) = "multiplicity notation" of m. "211" is read as 2^2 * 3^1 * 5^1 = 60.
PC(k) = A287352(k) = "pi-code" notation of k. Pi-code is a list of the first differences of indices 
        of prime divisors p of n, e.g., A287352(60) = 1,0,1,1 since 60 = 2 * 2 * 3 * 5. 
        The code is concatenated if all digits in the code are less than 10; 
        if not, the values are delimited by (.)."1011" is read as 2 * 2 * 3 * 5 = 60.
Both these notations serve to succinctly illustrate the prime decomposition of m and k respectively.
A054841 is useful for numbers that are products of small primes, while A287352 is useful for numbers
that are products of more widely separated primes.

n       m                       k     MN(m)   PC(m)
1	0	 	 	1	-	0
2	4	a	b	2	2	1
3	8	 	 	3	3	2
4	12	a	b	4	21	10
5	24	a	b	6	31	11
6	36	a	b	7	22	4
7	48	a	b	9	41	20
8	60	a	b	10	211	12
9	72	 	 	11	32	5
10	108	 	 	12	23	101
11	120	a	b	16	311	1000
12	180	a	b	20	221	102
13	240	a	b	22	411	14
14	336	 	 	24	4101	1001
15	360	a	b	27	321	200
16	480	 	 	28	511	103
17	504	 	 	30	3201	111
18	576	 	b	31	62	11
19	720	a	b	41	421	13
20	960	 	b	44	611	104
21	1080	 	 	48	331	10001
22	1260	a	b	49	2211	40
23	1440	 	 	61	521	18
24	1680	a	b	63	4111	202
25	2160	 	 	66	431	113
26	2520	a	b	76	3211	107
27	3360	 	 	83	5111	23
28	3780	 	 	87	2311	28
29	4320	 	 	93	531	29
30	5040	a	b	113	4211	30
31	7560	a	b	128	3311	1000000
32	10080	a	b	153	5211	205
33	12600	 	 	159	3221	2.14
34	15120	a	b	193	4311	44
35	20160	a	b	210	6211	1111
36	25200	a	b	223	4221	48
37	27720	a	b	229	32111	50
38	30240	 	 	262	5311	1.31
39	40320	 	 	281	7211	60
40	45360	a	b	291	4411	2.23
41	50400	a	b	325	5221	303
42	55440	a	b	327	42111	2.27
43	60480	 	b	352	6311	100004
44	75600	 	 	369	4321	2.0.11
45	83160	a	b	376	33111	1.0.0.14
46	90720	 	 	396	5411	10103
47	100800	 	b	428	6221	1.0.27
48	110880	a	b	477	52111	2.0.14
49	131040	 	 	486	521101	110000
50	151200	 	 	528	5321	100013
51	166320	a	b	543	43111	2.40
52	196560	 	 	561	431101	232
53	221760	a	b	623	62111	4.20
54	262080	 	 	637	621101	402
55	277200	a	b	670	42211	1.2.16
56	302400	 	 	686	6321	1300
57	332640	a	b	778	53111	1.76
58	443520	 	 	809	72111	140
59	498960	a	b	832	44111	1000005
60	524160	 	 	836	721101	1043
61	554400	a	b	942	52211	1.1.35
62	655200	 	 	951	522101	2.64
63	665280	a	b	1032	63111	1.0.0.1.12
64	831600	 	 	1092	43211	10122
65	982800	 	 	1096	432101	1.0.0.32
66	997920	 	 	1177	54111	5.23


Remarks and observations:

1. This analysis is based on terms 0 <= n <= 10^6.

2. Though m contains many terms of A002182, {1, 2, 6, 840, 720720, ...} are not found in m, and
   m contains {0, 8, 72, 108, 336, 480, 504, 576, 960, 1080, 1440, 2160, 3360, 3780, 4320, 12600, 
   30240, 40320, 60480, 75600, 90720, 100800, 131040, 151200, 196560, 262080, 302400, 443520, 524160, 
   655200, 831600, 982800, 997920, ...} that are not found in A002182.

Conjectures: (All excepting m(0) = 0.) 

1. Numbers in m are products of the first several consecutive primes or in the case of m(14), m(17),
   m(49), m(52), etc., the first primes that are consecutive but include one gap.

2. Outside of m(1), m(2), m(3), m(6), m(9), and m(10), the largest prime factor of m has multiplicity 1.

3. Outside of m(9), the multiplicities of prime factors p of m generally decrease or stay the same 
   as p increases.