# 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.