A294306 and indices of records in A294306.
Michael Thomas De Vlieger, St. Louis, Missouri, 201710271930, revised 201710291845.

Data range: 1 <= n <= 10^6.

The numbers i in A162306(n) divide n^k with k >= 0; these k are listed in row n of A280269.
T(n, m) = total number of each value that k takes in row n of A280269, 
with 0 <= m <= A280274(n).

Row n of A280269(10) = 0, 1, 2, 1, 3, 1, corresponding to A162306(10) = 1, 2, 4, 5, 8, 10,
since 1 | 10^0, 2 | 10^1, 4 | 10^2, 5 | 10^1, 8 | 10^3, and 10 | 10^1. 
There is 1 zero, 3 ones, 1 two, and 1 three.
thus A294306(10) = 1, 3, 1, 1. sum(A294306(10)) = A010846(10) = 6. 
Length of A294306(10) = A280274(10) + 1 = 4.

+-----------+
| Contents. |
+-----------+

Table 1. Population of values 0 <= k <= A280274(n) in A280269(n).
Table 2. Records T(n, m) and their positions n, m in A294306.
Table 3. Where a given value of m is maximum in A294306.
Table 4. A294306(A002110(n)) for 0 <= n <= 12.
Table 5. Smallest indices n where T(n, A280274(n)) does not equal 1.

+-------------------------------------------------------------------+
| Table 1. Population of values 0 <= k <= A280274(n) in A280269(n). |
+-------------------------------------------------------------------+

n = index
a = A294306(n) = T(n, m).
max(a) = largest value in T(n, m).
pos(max(a)) = index in row n of A294306 where record occurs.
tau(n) = A000005(n).
r(n) = A010846(n).
A280269(n) = row n in A280269 corresponding to A294306(n) and A162306(n).

  n     a = A294306(n)       max(a) pos(max(a))	tau(n) r(n)	A280269(n)
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  1	1                 	 1	0	 1	 1	0                                                                                                                                      
  2	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
  3	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
  4	1 2               	 2	1	 3	 3	0 1 1                                                                                                                                  
  5	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
  6	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
  7	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
  8	1 3               	 3	1	 4	 4	0 1 1 1                                                                                                                                
  9	1 2               	 2	1	 3	 3	0 1 1                                                                                                                                  
 10	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 11	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 12	1 5 2             	 5	1	 6	 8	0 1 1 1 1 2 2 1                                                                                                                        
 13	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 14	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 15	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 16	1 4               	 4	1	 5	 5	0 1 1 1 1                                                                                                                              
 17	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 18	1 5 2 1 1         	 5	1	 6	10	0 1 1 2 1 3 1 2 4 1                                                                                                                    
 19	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 20	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
 21	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 22	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
 23	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 24	1 7 3             	 7	1	 8	11	0 1 1 1 1 1 2 1 2 2 1                                                                                                                  
 25	1 2               	 2	1	 3	 3	0 1 1                                                                                                                                  
 26	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
 27	1 3               	 3	1	 4	 4	0 1 1 1                                                                                                                                
 28	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
 29	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 30	1 7 6 3 1         	 7	1	 8	18	0 1 1 2 1 1 3 2 1 2 1 4 2 2 3 2 3 1                                                                                                    
 31	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 32	1 5               	 5	1	 6	 6	0 1 1 1 1 1                                                                                                                            
 33	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 34	1 3 1 1 1 1       	 3	1	 4	 8	0 1 2 3 4 1 5 1                                                                                                                        
 35	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 36	1 8 4 1           	 8	1	 9	14	0 1 1 1 1 2 1 1 2 1 2 2 3 1                                                                                                            
 37	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 38	1 3 1 1 1 1       	 3	1	 4	 8	0 1 2 3 4 1 5 1                                                                                                                        
 39	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 40	1 7 3             	 7	1	 8	11	0 1 1 1 1 1 2 1 2 2 1                                                                                                                  
 41	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 42	1 7 6 3 1 1       	 7	1	 8	19	0 1 1 2 1 1 3 2 2 1 4 2 1 3 3 2 5 2 1                                                                                                  
 43	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 44	1 5 2 1           	 5	1	 6	 9	0 1 1 2 1 2 1 3 1                                                                                                                      
 45	1 5 2             	 5	1	 6	 8	0 1 1 1 1 2 2 1                                                                                                                        
 46	1 3 1 1 1 1       	 3	1	 4	 8	0 1 2 3 4 1 5 1                                                                                                                        
 47	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 48	1 9 4 1           	 9	1	10	15	0 1 1 1 1 1 2 1 1 2 1 3 2 2 1                                                                                                          
 49	1 2               	 2	1	 3	 3	0 1 1                                                                                                                                  
 50	1 5 2 2 1 1       	 5	1	 6	12	0 1 2 1 3 1 4 2 1 5 3 1                                                                                                                
 51	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 52	1 5 2 1           	 5	1	 6	 9	0 1 1 2 1 2 1 3 1                                                                                                                      
 53	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 54	1 7 3 2 2 1       	 7	1	 8	16	0 1 1 2 1 3 1 2 4 1 3 1 5 2 4 1                                                                                                        
 55	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 56	1 7 3             	 7	1	 8	11	0 1 1 1 1 1 2 1 2 2 1                                                                                                                  
 57	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 58	1 3 1 1 1 1       	 3	1	 4	 8	0 1 2 3 4 1 5 1                                                                                                                        
 59	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 60	1 11 11 3         	11	1	12	26	0 1 1 1 1 1 2 2 1 1 1 2 2 1 2 2 3 1 3 2 2 2 2 2 3 1                                                                                    
 61	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 62	1 3 1 1 1 1       	 3	1	 4	 8	0 1 2 3 4 1 5 1                                                                                                                        
 63	1 5 2             	 5	1	 6	 8	0 1 1 1 1 2 2 1                                                                                                                        
 64	1 6               	 6	1	 7	 7	0 1 1 1 1 1 1                                                                                                                          
 65	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 66	1 7 6 4 2 1 1     	 7	1	 8	22	0 1 1 2 1 3 2 1 2 4 2 1 3 3 5 1 2 2 4 3 6 1                                                                                            
 67	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 68	1 5 2 2           	 5	1	 6	10	0 1 1 2 2 1 3 1 3 1                                                                                                                    
 69	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
 70	1 7 6 3 1 1 1     	 7	1	 8	20	0 1 2 1 1 3 1 1 4 2 2 2 5 1 3 2 2 3 6 1                                                                                                
 71	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 72	1 11 6            	11	1	12	18	0 1 1 1 1 1 1 1 2 1 1 2 2 1 2 2 2 1                                                                                                    
 73	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 74	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
 75	1 5 2 1           	 5	1	 6	 9	0 1 1 2 1 1 3 2 1                                                                                                                      
 76	1 5 2 2           	 5	1	 6	10	0 1 1 2 2 1 3 1 3 1                                                                                                                    
 77	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 78	1 7 6 5 2 1 1     	 7	1	 8	23	0 1 1 2 1 3 2 2 1 4 2 3 1 3 5 2 1 4 2 3 6 3 1                                                                                          
 79	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 80	1 9 4             	 9	1	10	14	0 1 1 1 1 1 1 1 2 2 1 2 2 1                                                                                                            
 81	1 4               	 4	1	 5	 5	0 1 1 1 1                                                                                                                              
 82	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
 83	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 84	1 11 11 4 1       	11	1	12	28	0 1 1 1 1 1 2 2 1 1 2 2 1 2 3 1 3 2 1 2 2 3 2 2 3 2 4 1                                                                                
 85	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 86	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
 87	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
 88	1 7 3             	 7	1	 8	11	0 1 1 1 1 2 1 2 1 2 1                                                                                                                  
 89	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 90	1 11 11 4 3 1 1   	11	1	12	32	0 1 1 2 1 1 3 1 1 2 1 4 1 2 3 2 2 1 5 2 3 1 4 2 2 2 6 3 2 4 2 1                                                                        
 91	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 92	1 5 2 2           	 5	1	 6	10	0 1 1 2 2 1 3 1 3 1                                                                                                                    
 93	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
 94	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
 95	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
 96	1 11 5 2 1        	11	1	12	20	0 1 1 1 1 1 2 1 1 2 1 3 1 2 1 3 2 2 4 1                                                                                                
 97	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
 98	1 5 2 2 1 1 1     	 5	1	 6	13	0 1 2 1 3 1 4 2 5 1 3 6 1                                                                                                              
 99	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
100	1 8 4 2           	 8	1	 9	15	0 1 1 1 2 1 2 1 1 3 2 1 3 2 1                                                                                                          
101	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
102	1 7 6 5 3 2 1     	 7	1	 8	25	0 1 1 2 1 3 2 2 4 1 2 3 3 5 1 2 4 1 3 6 2 3 4 5 1                                                                                      
103	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
104	1 7 3             	 7	1	 8	11	0 1 1 1 1 2 1 2 1 2 1                                                                                                                  
105	1 7 6 1 1         	 7	1	 8	16	0 1 1 1 2 1 1 2 3 1 2 2 2 2 4 1                                                                                                        
106	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
107	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
108	1 11 6 3          	11	1	12	21	0 1 1 1 1 2 1 1 2 1 2 1 3 1 2 1 3 2 2 3 1                                                                                              
109	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
110	1 7 6 3 2 1 1     	 7	1	 8	21	0 1 2 1 3 1 1 4 2 1 2 5 3 2 2 1 6 4 3 2 1                                                                                              
111	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
112	1 9 4             	 9	1	10	14	0 1 1 1 1 1 1 1 2 2 1 2 2 1                                                                                                            
113	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
114	1 7 6 6 3 2 1     	 7	1	 8	26	0 1 1 2 1 3 2 2 4 2 1 3 3 5 2 1 4 3 1 6 3 2 4 5 3 1                                                                                    
115	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
116	1 5 2 2           	 5	1	 6	10	0 1 1 2 2 1 3 1 3 1                                                                                                                    
117	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
118	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
119	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
120	1 15 16 3 1       	16	2	16	36	0 1 1 1 1 1 1 2 1 1 1 2 2 1 1 2 3 1 2 2 1 2 2 2 3 1 2 2 2 2 4 2 2 2 3 1                                                                
121	1 2               	 2	1	 3	 3	0 1 1                                                                                                                                  
122	1 3 1 1 1 1 1     	 3	1	 4	 9	0 1 2 3 4 5 1 6 1                                                                                                                      
123	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
124	1 5 2 2           	 5	1	 6	10	0 1 1 2 2 1 3 1 3 1                                                                                                                    
125	1 3               	 3	1	 4	 4	0 1 1 1                                                                                                                                
126	1 11 11 4 3 2 1   	11	1	12	33	0 1 1 2 1 1 3 1 2 1 4 1 1 3 2 2 5 2 1 4 2 2 3 1 6 3 2 2 5 2 2 4 1                                                                      
127	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
128	1 7               	 7	1	 8	 8	0 1 1 1 1 1 1 1                                                                                                                        
129	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
130	1 7 6 4 2 1 1 1   	 7	1	 8	23	0 1 2 1 3 1 1 4 2 2 1 5 3 2 2 6 1 4 2 3 3 7 1                                                                                          
131	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
132	1 11 11 6 2       	11	1	12	31	0 1 1 1 1 2 2 1 1 2 2 1 2 3 3 1 2 1 2 3 3 1 2 4 2 3 2 3 2 4 1                                                                          
133	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
134	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
135	1 7 3 1           	 7	1	 8	12	0 1 1 1 1 2 1 1 2 2 3 1                                                                                                                
136	1 7 3 1           	 7	1	 8	12	0 1 1 1 2 1 2 1 2 1 3 1                                                                                                                
137	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
138	1 7 6 6 3 2 1 1   	 7	1	 8	27	0 1 1 2 1 3 2 2 4 2 1 3 3 5 2 1 4 3 6 1 3 4 2 5 3 7 1                                                                                  
139	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
140	1 11 11 3 1       	11	1	12	27	0 1 1 1 1 2 1 1 2 1 2 1 3 1 2 2 2 2 3 1 2 2 2 2 3 4 1                                                                                  
141	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
142	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
143	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
144	1 14 8            	14	1	15	23	0 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 2 1 2 2 2 2 1                                                                                          
145	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
146	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
147	1 5 2 1 1         	 5	1	 6	10	0 1 1 2 1 3 1 2 4 1                                                                                                                    
148	1 5 2 2 1         	 5	1	 6	11	0 1 1 2 2 3 1 3 1 4 1                                                                                                                  
149	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
150	1 11 11 9 5 2 1 1 	11	1	12	41	0 1 1 2 1 1 3 2 1 2 1 4 2 2 3 1 3 1 5 2 3 2 4 1 3 2 6 3 1 4 4 2 5 2 3 3 2 7 3 4 1                                                      
151	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
152	1 7 3 1           	 7	1	 8	12	0 1 1 1 2 1 2 1 2 1 3 1                                                                                                                
153	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
154	1 7 6 3 2 1 1 1   	 7	1	 8	22	0 1 2 1 3 1 1 4 1 2 5 2 2 3 6 1 3 2 4 2 7 1                                                                                            
155	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
156	1 11 11 6 2       	11	1	12	31	0 1 1 1 1 2 2 1 1 2 2 2 1 3 3 2 1 2 1 3 3 2 1 4 3 2 3 2 4 2 1                                                                          
157	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
158	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
159	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
160	1 11 5 1          	11	1	12	18	0 1 1 1 1 1 1 1 2 1 1 2 2 1 2 3 2 1                                                                                                    
161	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
162	1 9 4 3 3 2 1 1   	 9	1	10	24	0 1 1 2 1 3 1 2 4 1 3 1 5 2 4 1 6 3 1 5 2 7 4 1                                                                                        
163	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
164	1 5 2 2 1         	 5	1	 6	11	0 1 1 2 2 3 1 3 1 4 1                                                                                                                  
165	1 7 6 3 1         	 7	1	 8	18	0 1 1 2 1 1 2 3 1 2 1 2 4 2 2 3 3 1                                                                                                    
166	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
167	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
168	1 15 16 4 2       	16	2	16	38	0 1 1 1 1 1 1 2 1 1 2 2 1 1 3 1 2 2 1 2 2 3 1 2 2 2 4 1 2 2 3 2 2 3 2 2 4 1                                                            
169	1 2               	 2	1	 3	 3	0 1 1                                                                                                                                  
170	1 7 6 4 2 2 1 1   	 7	1	 8	24	0 1 2 1 3 1 4 1 2 2 5 1 3 2 6 2 4 1 2 3 7 3 5 1                                                                                        
171	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
172	1 5 2 2 1         	 5	1	 6	11	0 1 1 2 2 3 1 3 1 4 1                                                                                                                  
173	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
174	1 7 6 6 5 2 1 1   	 7	1	 8	29	0 1 1 2 1 3 2 2 4 2 3 3 1 5 2 4 3 1 6 3 4 1 5 3 2 7 4 4 1                                                                              
175	1 5 2             	 5	1	 6	 8	0 1 1 1 1 2 2 1                                                                                                                        
176	1 9 4             	 9	1	10	14	0 1 1 1 1 1 1 2 1 2 1 2 2 1                                                                                                            
177	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
178	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
179	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
180	1 17 20 5 1       	20	2	18	44	0 1 1 1 1 1 2 1 1 1 1 2 1 1 2 2 2 1 3 1 2 1 2 2 2 1 3 2 2 2 2 1 3 2 2 2 3 4 2 2 2 3 2 1                                                
181	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
182	1 7 6 3 2 1 1 1   	 7	1	 8	22	0 1 2 1 3 1 1 4 1 2 5 2 2 3 6 1 2 3 4 7 2 1                                                                                            
183	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
184	1 7 3 1           	 7	1	 8	12	0 1 1 1 2 1 2 1 2 1 3 1                                                                                                                
185	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
186	1 7 6 6 5 2 1 1   	 7	1	 8	29	0 1 1 2 1 3 2 2 4 2 3 3 1 5 2 4 3 1 6 3 4 1 5 3 2 7 4 4 1                                                                              
187	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
188	1 5 2 2 1         	 5	1	 6	11	0 1 1 2 2 3 1 3 1 4 1                                                                                                                  
189	1 7 3             	 7	1	 8	11	0 1 1 1 1 1 2 1 2 2 1                                                                                                                  
190	1 7 6 4 2 2 1 1   	 7	1	 8	24	0 1 2 1 3 1 4 1 2 2 5 1 3 2 6 2 4 1 2 3 7 3 5 1                                                                                        
191	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
192	1 13 6 3 2        	13	1	14	25	0 1 1 1 1 1 2 1 1 2 1 3 1 2 1 3 1 2 4 1 3 2 2 4 1                                                                                      
193	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
194	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
195	1 7 6 3 1         	 7	1	 8	18	0 1 1 2 1 1 2 3 1 2 1 2 4 2 3 3 2 1                                                                                                    
196	1 8 4 2 1         	 8	1	 9	16	0 1 1 1 2 1 2 1 3 1 2 3 1 2 4 1                                                                                                        
197	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
198	1 11 11 4 4 2 2 1 	11	1	12	36	0 1 1 2 1 3 1 1 2 4 1 1 3 2 5 1 2 2 4 2 6 1 3 2 3 5 1 2 2 7 2 4 2 4 6 1                                                                
199	1 1               	 1	0	 2	 2	0 1                                                                                                                                    
200	1 11 6 1          	11	1	12	19	0 1 1 1 1 1 2 1 1 2 1 1 2 2 1 2 3 2 1                                                                                                  
201	1 3 1 1 1         	 3	1	 4	 7	0 1 2 3 1 4 1                                                                                                                          
202	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
203	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
204	1 11 11 7 3       	11	1	12	33	0 1 1 1 1 2 2 1 2 1 2 2 3 3 1 2 2 1 3 3 1 2 4 3 1 3 4 2 2 2 4 3 1                                                                      
205	1 3 1 1           	 3	1	 4	 6	0 1 2 1 3 1                                                                                                                            
206	1 3 1 1 1 1 1 1   	 3	1	 4	10	0 1 2 3 4 5 6 1 7 1                                                                                                                    
207	1 5 2             	 5	1	 6	 8	0 1 1 1 2 1 2 1                                                                                                                        
208	1 9 4             	 9	1	10	14	0 1 1 1 1 1 1 2 1 2 1 2 2 1                                                                                                            
209	1 3 1             	 3	1	 4	 5	0 1 1 2 1                                                                                                                              
210	1 15 25 14 7 3 2 1	25	2	16	68	0 1 1 2 1 1 1 3 2 1 2 1 1 4 2 2 1 3 2 3 2 1 5 1 2 3 1 2 4 2 2 3 3 2 2 6 1 3 2 4 4 2 2 5 2 2 1 3 4 3 3 2 7 3 2 4 2 2 5 4 3 2 2 3 6 2 3 1

////// Observations //////

1. Row 1 = 1 and T(n, 0) = 1 for all n, since 1 is the empty product and divides n^0.
2. Row p prime = 1, 1, since the only divisors of p are 1 and p; 1 | p^0, and p | p^1.
3. Row p^e = 1, e, since the only numbers in A162306(p^e) are 1 and p^k for 1 <= k <= e.
4. Row length of a(n) > 2 for n with omega(n) > 1.
5. Row length = A280274(n) + 1.
6. Total of row n = A010846(n).
7. Sum of terms m = {0, 1} in row n = A000005(n).
8. Terms in row n of A294306 start at 1, generally quickly rise to a maximum, then gradually decline to 1 at
   m = A280274(n).
9. See Table 5 for the smallest n that have last terms in row n of A294306 that are larger than 1.

+---------------------------------------------------------------+
| Table 2. Records T(n, m) and their positions n, m in A294306. |
+---------------------------------------------------------------+

j = Index of this table.
n = Index of record T(n, m) in A294306.
m = Smallest term in row n of A294306 that sets a record in that sequence.
tau(n) =  A000005(n).
r(n) =    A010846(n).
A2182 =   index of n in A002182.
A244052 = index of n in A244052.
A293555 = index of n in A293555.
MN(n) = rev(A054841(n)), little-endian concatenation of exponents e of prime divisors of n.
    example: the number 84 = 2^2 * 3 * 7, thus MN(84) = 2101.

 j            n m      T(n, m) tau(n)     r(n) A2182 A244052 A293555 A288784    MN(n)
----------------------------------------------------------------------------------------------
 1            1	0	   1	  1	    1	 1	  1	  1	  1	0             
 2            4	1	   2	  3	    3	 3	  3	   	  4	2             
 3            6	1	   3	  4	    5	 4	  4	  2	  5	11            
 4           12	1	   5	  6	    8	 5	  6	   	  7	21            
 5           24	1	   7	  8	   11	 6	  8	   	  9	31            
 6           36	1	   8	  9	   14	 7	   	   	   	22            
 7           48	1	   9	 10	   15	 8	   	   	   	41            
 8           60	1	  11	 12	   26	 9	 11	  7	 12	211           
 9          120	2	  16	 16	   36	10	 14	   	 15	311           
10          180	2	  20	 18	   44	11	 16	   	 17	221           
11          210	2	  25	 16	   68	  	 17	 13	 18	1111          
12          360	2	  29	 24	   58	13	   	   	   	321           
13          420	2	  44	 24	   96	  	 20	 16	 21	2111          
14          840	2	  63	 32	  131	15	 22	 18	 23	3111          
15         1260	2	  77	 36	  156	16	 24	   	 25	2211          
16         1680	2	  82	 40	  174	17	 26	   	 27	4111          
17         2310	2	  90	 32	  283	  	 29	 24	 30	11111         
18         2520	2	 110	 48	  206	18	   	   	   	3211          
19         4620	2	 155	 48	  382	  	 33	 28	 35	21111         
20         7560	2	 157	 64	  308	20	   	   	   	3311          
21         9240	2	 220	 64	  505	  	 37	 32	 39	31111         
22        13860	2	 266	 72	  593	  	 41	 36	 43	22111         
23        18480	2	 285	 80	  660	  	 43	   	 45	41111         
24        27720	2	 377	 96	  766	25	 47	   	 49	32111         
25        55440	2	 488	120	  979	28	   	   	   	42111         
26        60060	2	 512	 96	 1526	  	 54	 48	 56	211111        
27        83160	2	 534	128	 1124	29	   	   	   	33111         
28       110880	2	 599	144	 1238	30	   	   	   	52111         
29       120120	2	 723	128	 1977	  	 59	 53	 61	311111        
30       180180	2	 869	144	 2294	  	 61	 55	 63	221111        
31       240240	2	 934	160	 2538	  	 63	 57	 65	411111        
32       360360	2	1226	192	 2921	  	 67	   	 69	321111        
33       690690	3	1244	128	 5119	  	 74	 67	 76	111111001     
34	 720720	2	1583	240	 3689	38	   	  	   	421111     
35	1021020	3	1963	192	 6225	  	 78	71	 83	2111111    
36	1141140	3	1986	192	 6337	  	 79	72	 84	21111101   
37	1381380	3	2031	192	 6546	  	 80	73	 85	211111001  
38	1741740	3	2090	192	 6828	  	   	  	   	2111110001 
39	1861860	3	2112	192	 6917	  	   	  	   	21111100001
40*	2042040	3	2621	256	 7928	  	 84	77	 89	3111111    
41	2282280	3	2650	256	 8055	  	 85	78	 90	31111101   
42	2762760	3	2711	256	 8294	  	   	  	   	311111001  
43	3063060	3	3153	288	 9099	  	 88	81	 93	2211111    
44	3423420	3	3192	288	 9236	  	 89	82	 94	22111101   
45	4084080	3	3271	320	10010	  	 92	85	 97	4111111    
46	4564560	3	3304	320	10155	  	 93	86	 98	41111101   
47	5105100	3	3307	288	10777	  	 95	88	100	2121111    
48	5225220	3	3354	288	 9847	  	   	  	   	2211110001 
49	5525520	3	3374	320	10428	  	   	  	   	411111001  
50	5585580	3	3383	288	 9958	  	   	  	   	22111100001
51	6126120	3	4217	384	11441	  	 97	90	102	3211111    
52	6846840	3	4264	384	11589	  	   	  	   	32111101   
53	8288280	3	4348	384	11879	  	   	  	   	321111001  
54	9189180	3	4398	384	13037	  	103	97	108	2311111    
55	9699690	3	4771	256	19985	  	104	98	109	11111111   

////// Observations //////

1. n < 1741740 is either highly composite or highly regular, i.e., in A002182 or A244052, sometimes both.
2. n = 1 pertains to the empty product 1.
3. n sets a record in A294306 at m-th term for 0 <= m <= 1 for j <= 8. Such a value of m pertains
   to a divisor of n.
4. The only observed squarefree n in this sequence are {1, 6, 210, 2310, 690690}.
5. Asterisk in first column denotes the first conjectural term, based on assumption of observation 1.
6. Double asterisk denotes the first conjectural index in A293555.

////// Conjectures //////

1. n sets records at m > 2. (Observed: For j = 33, n = 690690, m = 3, 201710282230).
2. n is either highly composite or highly regular, i.e., in A002182 or A244052, sometimes both. 
   (proved FALSE by term 1741740, 201710291400).
3. n is a product of the smallest omega(n) primes. 
   (proved FALSE by term 690690, 201710282230).

+----------------------------------------------------------+
| Table 3. Where a given value of m is maximum in A294306. |
+----------------------------------------------------------+

m = exponent of n such that more numbers in A162306(n) divide n^m than any other 
    nonzero positive exponent. 
n = row in A294306 where T(n, m) is greatest at m.
PC(n) = A287352(n) = pi(lpf(n)) followed by first differences of indices of the rest
    of the prime divisors of n, delimited by ".". Example: PC(84) = 1.0.1.2
T(m,n) = max(A294306(n)).
A294306(n) = population of each value 0..A280274(n) in A280269(n). 

m	     n  PC(n) 	       T(m,n)   A294306(n)
------------------------------------------------------------------
0	     1	0       	 1	1                                      
1	     4	1.0      	 2	1  2                                    
2	   120	1.0.0.1.1   	16	1 15 16  3 1                            
3	   876	1.0.1.19	12	1 11 11 12 8  3  1                        
4	   654	1.1.27  	 8	1  7  6  7 8  5  3  2  1  1                    
5	  3918	1.1.117 	10	1  7  6  7 9 10  7  5  3  2  2 1               
6	 23334	1.1.537 	12	1  7  6  7 9 11 12  9  7  5  3 3 2 1 1        
7	139998	1.1.2600	14	1  7  6  7 9 11 13 14 11  9  7 4 4 3 2 2 1 1
8	839814	1.1.13004	16	1  7  6  7 9 11 13 15 16 13 11 9 6 5 4 3 3 2 2 1

////// Observations //////

For m >= 2, omega(n) = 3.

+-------------------------------+
| Table 4. A294306(A002110(n)). |
+-------------------------------+

(Chart rotated counterclockwise for concision)
x axis = primorial A002110(x)
y axis = terms of A294306(A002110(x)), with m = 0 at bottom.
 	 	 	 	  	  	   	    	    	     	     	      	      1
 	 	 	 	  	  	   	    	    	     	     	      	      2
 	 	 	 	  	  	   	    	    	     	     	      	      3
 	 	 	 	  	  	   	    	    	     	     	      	      7
 	 	 	 	  	  	   	    	    	     	     	      	     13
 	 	 	 	  	  	   	    	    	     	     	     1	     23
 	 	 	 	  	  	   	    	    	     	     	     1	     38
 	 	 	 	  	  	   	    	    	     	     	     3	     63
 	 	 	 	  	  	   	    	    	     	     	     6	    103
 	 	 	 	  	  	   	    	    	     	     	    12	    163
 	 	 	 	  	  	   	    	    	     	    1	    20	    245
 	 	 	 	  	  	   	    	    	     	    2	    34	    367
 	 	 	 	  	  	   	    	    	     	    3	    54	    546
 	 	 	 	  	  	   	    	    	     	    6	    88	    789
 	 	 	 	  	  	   	    	    	     	   12	   138	   1132
 	 	 	 	  	  	   	    	    	    1	   19	   205	   1599
 	 	 	 	  	  	   	    	    	    2	   33	   307	   2232
 	 	 	 	  	  	   	    	    	    3	   53	   449	   3093
 	 	 	 	  	  	   	    	    	    7	   85	   649	   4235
 	 	 	 	  	  	   	    	   1	   13	  128	   917	   5750
 	 	 	 	  	  	   	    	   1	   20	  188	  1285	   7751
 	 	 	 	  	  	   	    	   2	   33	  276	  1784	  10399
 	 	 	 	  	  	   	    	   5	   53	  401	  2457	  13857
 	 	 	 	  	  	   	    	   9	   82	  570	  3350	  18383
 	 	 	 	  	  	   	   1	  15	  121	  801	  4537	  24299
 	 	 	 	  	  	   	   2	  23	  177	 1116	  6125	  32024
 	 	 	 	  	  	   	   4	  39	  261	 1541	  8229	  42137
 	 	 	 	  	  	   	   8	  59	  370	 2122	 11021	  55362
 	 	 	 	  	  	  1	  12	  88	  524	 2906	 14741	  72691
 	 	 	 	  	  	  2	  19	 129	  738	 3961	 19690	  95495
 	 	 	 	  	  	  4	  31	 190	 1039	 5409	 26326	 125588
 	 	 	 	  	 1	  7	  49	 279	 1457	 7368	 35204	 165497
 	 	 	 	  	 1	 11	  73	 398	 2035	10064	 47225	 218744
 	 	 	 	  	 2	 18	 110	 578	 2849	13768	 63519	 290168
 	 	 	 	  	 5	 29	 166	 834	 4005	18916	 85848	 386806
 	 	 	 	 1	 8	 47	 249	1208	 5643	26109	116634	 518484
 	 	 	 	 2	14	 74	 378	1757	 8013	36287	159410	 698018
 	 	 	 	 3	22	118	 573	2576	11432	50483	216990	 929231
 	 	 	1	 7	40	193	 862	3742	15942	67304	277412	1137032
 	 	 	3	14	68	292	1212	4771	18412	70213	263213	 979166
 	 	1	6	25	90	301	 966	3025	 9330	28501	 86526	 261625
 	1	3	7	15	31	 63	 127	 255	  511	 1023	  2047	   4095
1	1	1	1	 1	 1	  1	   1	   1	    1	    1	     1	      1

0	1	2	3	 4	 5	  6	   7	   8	    9	   10	    11	     12

						A002110(x)

+--------------------------------------------------------------------+
| Table 5. Smallest indices where T(n, A280274(n)) does not equal 1. |
+--------------------------------------------------------------------+

s = T(n, A280274(n))
n = smallest n for which s is the last term in A294306.
MN(n) = rev(A054841(n)), little-endian concatenation of exponents e of prime divisors of n.
    example: the number 84 = 2^2 * 3 * 7, thus MN(84) = 2101.

 s           n  A294306(n)      MN(n)
------------------------------------------------------------------------
 1	      1	1               	0                       
 2	      4	1 2             	2                       
 3	      8	1 3             	3                       
 4	     16	1 4             	4                       
 5	     32	1 5             	5                       
 6	     64	1 6             	6                       
 7	    128	1 7             	7                       
 8	    144	1 14 8          	4.2                     
 9	    512	1 9             	9                       
10	    800	1 17 10         	5.0.2                   
11	   2016	1 35 47 11      	5.2.0.1                 
12	    432	1 19 12         	4.3                     
13	   8192	1 13            	13                      
14	   6272	1 23 14         	7.0.0.2                 
15	    864	1 23 15         	5.3                     
16	  12544	1 26 16         	8.0.0.2                 
17	  39744	1 55 81 17      	(2^6 * 3^3 * 23^1)      
18	   1728	1 27 18         	6.3                     
19	  37152	1 47 68 19      	(2^5 * 3^3 * 43^1)      
20	  40608	1 47 68 20      	(2^5 * 3^3 * 47^1)      
21	  16000	1 31 21         	7.0.3                   
22	  52704	1 47 68 22      	(2^5 * 3^3 * 61^1)      
23	  21600	1 71 121 23     	5.3.2                   
24	   5184	1 34 24         	6.4                     
25	 368064	1 69 106 25     	(2^6 * 3^4 * 71^1)      
26	  42336	1 71 121 26     	5.3.0.2                 
27	  64000	1 39 27         	9.0.3                   
28	  10368	1 39 28         	7.4                     
29	 392000	1 83 144 29     	6.0.3.2                 
30	 351232	1 43 30         	10.0.0.3                
31	1316736	1 79 123 31     	(2^7 * 3^4 * 127^1)     
32	 784000	1 95 167 32     	7.0.3.2                 
33	 702464	1 47 33         	11.0.0.3                
34	1534464	1 99 157 34     	(2^9 * 3^4 * 37^1)      
35	  31104	1 47 35         	7.5                     
36	 320000	1 49 36         	9.0.4                   
39	1274400	1 143 434 193 39	(2^5 * 3^3 * 5^2 * 59^1)
40	  62208	1 53 40         	8.5                     
42	1254528	1 119 218 42    	7.4.0.0.2               
43	1533600	1 143 434 202 43	(2^5 * 3^3 * 5^2 * 71^1)
44	1280000	1 59 44         	11.0.4                  
45	 124416	1 59 45         	9.5                     
54	 373248	1 69 54         	9.6                     
59	1296000	1 159 313 59    	7.4.3                   
60	 746496	1 76 60         	10.6                    
66	1492992	1 83 66         	11.6                    

////// Observations //////

1. The last term m = A280274(n) in row n of A294306 is often 1: for 1 <= n <= 10^6, this is
   the case in all but 166235 instances.
2. Among n we see 1 and the powers {2, 3, 4, 5, 6, 7, 9} of 2, and the powers {2, 3} of 12.

////// Conjecture //////

1. s may have any positive value.

+------------------+
| Revision Record. |
+------------------+

201710271930 created, data range 1 <= n <= A002110(6) = 30030.
201710281545 revised, data range 1 <= n <= A002110(7) = 510510, tables 2 - 5 added.
201710282230 revised, data range 1 <= n <= 10^6.
201710290845 revised, conjectural extension of Table 2.
201710291400 revised, conjecture disproven.
201710291845 revised, data range 1 <= n <= 2042040.

+------------------+
| Files available. |
+------------------+

b280269-p7.txt        510510 rows,   50,316,476 terms   151,378 kb
b280269-6.txt           10^6 rows,  115,171,136 terms   347,096 kb
b280269-3111111.txt  2042040 rows,  276,124,541 terms   833,821 kb

b280274-6.txt           10^6 terms                        4,000 kb

b294306-p7.txt        510510 rows,    5,016,703 terms    16,923 kb
b294306-6.txt           10^6 rows,   10,368,303 terms    35,035 kb 
b294306-3111111.txt  2042040 rows,   22,194,260 terms    75,216 kb

b294306r-6.txt          10^6 terms                       10,180 kb
b294306r-3111111.txt 2042040 terms                       21,978 kb

(eof)