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) --------------------------------------------------------------------------------------------- 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)