# Multiplicities of primes for FactorInteger[A244052(n)]
# by Michael T. De Vlieger, 2 July 2014, revised 10 February 2015
# with data after term 104 by David A. Corneth.
#
#				Multiplicity of Prime (one prime per column)
# Primorial		    	    1111223344
n p#	   A244052     A244053	23571379391713
1	         1	     1	0
2 2#		 2	     2	1
3	         4	     3	2
4 3#	         6	     5	11
5	        10	     6	101
6	        12	     8	21
7	        18	    10	12
8	        24	    11	31
9  5#	        30	    18	111
10	        42	    19	1101
11	        60	    26	211
12	        84	    28	2101
13	        90	    32	121
14	       120	    36	311
15	       150	    41	112
16	       180	    44	221
17 7#	       210	    68	1111
18	       330	    77	11101
19	       390	    80	111001
20	       420	    96	2111
21	       630	   115	1211
22	       840	   131	3111
23	      1050	   145	1121
24	      1260	   156	2211
25	      1470	   166	1112
26	      1680	   174	4111
27	      1890	   183	1311
28	      2100	   192	2121
29 11#	      2310	   283	11111
30	      2730	   295	111101
31	      3570	   313	1111001
32	      3990	   322	11110001
33	      4620	   382	21111
34	      5460	   395	211101
35	      6930	   452	12111
36	      8190	   463	121101
37	      9240	   505	31111
38	     10920	   519	311101
39	     11550	   551	11211
40	     13650	   567	112101
41	     13860	   593	22111
42	     16170	   629	11121
43	     18480	   660	41111
44	     20790	   691	13111
45	     23100	   717	21211
46	     25410	   743	11112
47 	     27720	   766	32111
48 13#	     30030	  1161	111111
49	     39270	  1224	1111101
50	     43890	  1253	11111001
51	     46410	  1257	1111011
52	     51870	  1285	11110101
53	     53130	  1306	111110001
54	     60060	  1526	211111
55	     78540	  1597	2111101		# <-- Note 1.
56	     87780	  1631	21111001
57	     90090	  1779	121111
58	    117810	  1856	1211101
59	    120120	  1977	311111
60	    150150	  2144	112111
61	    180180	  2294	221111
62	    210210	  2420	111211
63	    240240	  2538	311111
64	    270270	  2645	131111
65	    300300	  2743	212111
66	    330330	  2836	111121
67	    360360	  2921	321111
68	    390390	  3001	111112
69	    420420	  3080	211211
70	    450450	  3153	122111
71	    480480	  3223	511111
72 17#	    510510	  4843	1111111
73	    570570	  4939	11111101
74	    690690	  5119	111111001
75	    746130	  5138	11111011
76	    870870	  5364	1111110001
77	    930930	  5436	11111100001
78	   1021020	  6225	2111111
79	   1141140	  6337	21111101
80	   1381380	  6546	211111001
81	   1492260	  6560	21111011
82	   1531530	  7178	1211111
83	   1711710	  7299	12111101
84	   2042040	  7928	3111111
85	   2282280	  8055	31111101
86	   2552550	  8553	1121111		# <-- Note 2.
87	   2852850	  8685	11211101
88	   3063060	  9099	2211111
89	   3423420	  9236	22111101
90	   3573570	  9580	1112111
91	   3993990	  9719	11121101
92	   4084080	 10010	4111111
93	   4564560	 10155	41111101
94	   4594590	 10414	1311111
95	   5105100	 10777	2121111
96	   5615610	 11120	1111211
97	   6126120	 11441	3211111
98	   6636630	 11740	1111121
99	   7147140	 12027	2112111
100	   7657650	 12293	1221111
101	   8168160	 12549	5111111
102	   8678670	 12799	1111112
103	   9189180 	 13037	2311111
104 19#	   9699690	 19985	11111111	# <-- Note 3.
105	  11741730	 20605	111111101
106	  13123110	 20929	111111011
107	  14804790	 21453	1111111001
108	  15825810	 21713	11111110001
109	  16546530	 21769	1111110101
110	  17687670	 22028	11111101001
111	  18888870	 22443	111111100001
112	  19399380	 25289	21111111
113	  23483460	 26005	211111101
114	  26246220	 26370	2111110101
115	  29099070	 28924	12111111
116	  35225190	 29701	121111101
117	  38798760	 31776	31111111
118	  46966920	 32594	311111101
119	  48498450	 34150	11211111
120	  58198140	 36204	22111111
121	  67897830	 38028	11121111
122	  77597520	 39660	41111111
123	  87297210	 41161	13111111
124	  96996900	 42543	21211111
125	 106696590	 43827	11112111
126	 116396280	 45029	31211111
127	 126095970	 46156	11111211
128	 135795660	 47233	21121111
129	 145495350	 48240	12211111
130	 155195040	 49202	51111111
131	 164894730	 50130	11111121
132	 174594420	 51014	23111111
133	 184294110	 51861	11111112
134	 193993800	 52680	31211111
135	 203693490	 53468	12121111
136	 213393180	 54226	21112111
137 23#	 223092870	 83074	111111111
138	 281291010	 86054	1111111101
139	 300690390	 86978	11111111001
140	 340510170	 88168	1111111011
141	 358888530	 89598	111111110001
142	 397687290	 91214	1111111100001
143	 417086670	 91993	11111111000001
144	 446185740	103747	211111111
145	 562582020	107188	2111111101
146	 601380780	108267	21111111001
147	 669278610	117837	121111111
148	 843873030	121572	1211111101
149	 892371480	128844	311111111
150	1115464350		112111111	# <-- Note 4.
151	1338557220		221111111
152	1561650090		111211111
153	1784742960		411111111
154	2007835830		131111111
155	2230928700		212111111
156	2454021570		111121111
157	2677114440		321111111
158	2900207310		111112111
159	3123300180		211211111
160	3346393050		122111111
161	3569485920		511111111
162	3792578790		111111211
163	4015671660		231111111
164	4238764530		111111121
165	4461857400		312111111
166	4684950270		121211111
167	4908043140		211121111
168	5131136010		111111112
169	5354228880		421111111
170	5577321750		113111111
171	5800414620		211112111
172	6023507490		141111111
173	6246600360		311211111
174 29#	6469693230		1111111111
175	6915878970		11111111101
176	8254436190		111111111001
177	8720021310		11111111011	# <-- Note 5.
178	9146807670		1111111110001
179	9592993410		11111111100001
180	10485364890		111111111000001
181	11823922110		1111111110000001
182	12939386460		2111111111
183	13831757940		21111111101
184	16508872380		211111111001
185	17440042620		21111111011	# <-- Note 5.
186	18293615340		2111111110001
187	19409079690		1211111111
188	20747636910		12111111101
189	24763308570		121111111001
190	25878772920		3111111111
191	27663515880		31111111101
192	32348466150		1121111111
193	34579394850		11211111101
194	38818159380		2211111111
195	41495273820		22111111101
196	45287852610		1112111111
197	48411152790		11121111101
198	51757545840		4111111111
199	55327031760		41111111101
200	58227239070		1311111111
201	62242910730		13111111101
202	64696932300		2121111111
203	69158789700		21211111101
204	71166625530		1111211111
205	76074668670		11112111101
206	77636318760		3211111111
207	82990547640		32111111101
208	84106011990		1111121111
209	89906426610		11111211101
210	90575705220		2112111111
211	96822305580		21121111101
212	97045398450		1221111111
213	103515091680		5111111111
214	109984784910		1111112111
215	116454478140		2311111111
216	122924171370		1111111211
217	129393864600		3121111111
218	135863557830		1212111111
219	142333251060		2111211111
220	148802944290		1111111121
221	155272637520		4211111111
222	161742330750		1131111111
223	168212023980		2111121111
224	174681717210		1411111111
225	181151410440		3112111111
226	187621103670		1111111112
227	194090796900		2221111111
228 31#	200560490130		11111111111


# Note 1. Conjecture: a244052(n) is set by products of primorial pi(n)#,
# then products of pi(n - 1)# pi(n + 1),
# pi(n - 1)#pi(n + 1)pi(n + 2), even some products of pi(n - 2)#pi(n)pi(n + 1),
# etc. until these products exceed multiples k(pi(n)#), with 2 <= k < pi(n + 1).
# Products tested to find 54 < n < 86:
# DeleteDuplicates[
# Sort[Flatten[
#   KroneckerProduct[Range[2310, 30030, 210], 
#    Apply[Times, Rest[Subsets[{11, 13, 17, 19}]], {1}]]]]]
# Note 2. Ensuing values consider only multiples k of primorial pi(n)#
# and any k(pi(n - 1))# * pi(n + 1) > k(pi(n)#) or k(pi(n - 1)#) * pi(n + 2) > k(pi(n)#)
# Note 3. Terms 105-149 of A244052 were projected 16 July 2014, but not tested until 
# David Corneth calculated them 9 February 2015. 
# (Terms skipped were 109, 110, 114, 148.) 
# David Corneth computed terms 105-149 of A244053 on 9 February.
# Note 4. Terms following 149 were projected 9-10 February 2015 
# based on the patterns in the preceding primorial groups.