Records A293556 and their indices A293555 in A243822. Michael Thomas De Vlieger, St. Louis, Missouri, 201710101400, updated 201711171130. Contents: 1. Table 1: Records in A243822 and their indices. 2. Observations regarding Table 1. 3. Table 2: Appearance of terms of A244052 among records of A243822. 4. Observations regarding Table 2. +------------------------------------------------+ | Table 1: Records in A243822 and their indices. | +------------------------------------------------+ n = index in this document. A293555(n) = index of record r in A243822. k = index of record A293556(n) in A244052. s = A243822(n) = record in A243822. tau(n) = A000005(n). A293556(n) = A010846(n) = A243822(n) + A000005(n). MN(n) = multiplicity notation = rev(A054841(n)), shorthand notation that lists exponents e of p, pi(p) conveyed by position from left. e.g., 60 = 2^2 * 3 * 5 = "211". n A293555(n) k s(n) tau(n) A293556(n) MN(n) -------------------------------------------------------------------- 1 1 1 0 1 1 0 2 6 4 1 4 5 11 3 10 5 2 4 6 101 4 18 7 4 6 10 12 5 30 9 10 8 18 111 6 42 10 11 8 19 1101 7 60 11 14 12 26 211 8 78 15 8 23 110001 9 84 12 16 12 28 2101 10 90 13 20 12 32 121 11 126 21 12 33 1201 12 150 15 29 12 41 112 13 210 17 52 16 68 1111 14 330 18 61 16 77 11101 15 390 19 64 16 80 111001 16 420 20 72 24 96 2111 17 630 21 91 24 115 1211 18 840 22 99 32 131 3111 19 990 100 24 124 12101 20 1050 23 121 24 145 1121 21 1470 25 142 24 166 1112 22 1890 27 151 32 183 1311 23 2100 28 156 36 192 2121 24 2310 29 251 32 283 11111 25 2730 30 263 32 295 111101 26 3570 31 281 32 313 1111001 27 3990 32 290 32 322 11110001 28 4620 33 334 48 382 21111 29 5460 34 347 48 395 211101 30 6930 35 404 48 452 12111 31 8190 36 415 48 463 121101 32 9240 37 441 64 505 31111 33 10920 38 455 64 519 311101 34 11550 39 503 48 551 11211 35 13650 40 519 48 567 112101 36 13860 41 521 72 593 22111 37 16170 42 581 48 629 11121 38 19110 595 48 643 111201 39 20790 44 627 64 691 13111 40 23100 45 645 72 717 21211 41 25410 46 695 48 743 11112 42 30030 48 1097 64 1161 111111 43 39270 49 1160 64 1224 1111101 44 43890 50 1189 64 1253 11111001 45 46410 51 1193 64 1257 1111011 46 51870 52 1221 64 1285 11110101 47 53130 53 1242 64 1306 111110001 48 60060 54 1430 96 1526 211111 49 78540 55 1501 96 1597 2111101 50 87780 56 1535 96 1631 21111001 51 90090 57 1683 96 1779 121111 52 117810 58 1760 96 1856 1211101 53 120120 59 1849 128 1977 311111 54 150150 60 2048 96 2144 112111 55 180180 61 2150 144 2294 221111 56 210210 62 2324 96 2420 111211 57 240240 63 2378 160 2538 411111 58 270270 64 2517 128 2645 131111 59 300300 65 2599 144 2743 212111 60 330330 66 2740 96 2836 111121 61 390390 68 2905 96 3001 111112 62 420420 69 2936 144 3080 211211 63 450450 70 3009 144 3153 122111 64 480480 71 3031 192 3223 511111 65 510510 72 4715 128 4843 1111111 66 570570 73 4811 128 4939 11111101 67 690690 74 4991 128 5119 111111001 68 746130 75 5010 128 5138 11111011 69 870870 76 5236 128 5364 1111110001 70 930930 77 5308 128 5436 11111100001 71 1021020 78 6033 192 6225 2111111 72 1141140 79 6145 192 6337 21111101 73 1381380 80 6354 192 6546 211111001 74 1492260 81 6368 192 6560 21111011 75 1531530 82 6986 192 7178 1211111 76 1711710 83 7107 192 7299 12111101 77 2042040 84 7672 256 7928 3111111 78 2282280 85 7799 256 8055 31111101 79 2552550 86 8361 192 8553 1121111 80 2852850 87 8493 192 8685 11211101 81 3063060 88 8811 288 9099 2211111 82 3423420 89 8948 288 9236 22111101 83 3573570 90 9388 192 9580 1112111 84 3993990 91 9527 192 9719 11121101 85 4084080 92 9690 320 10010 4111111 86 4564560 93 9835 320 10155 41111101 87 4594590 94 10158 256 10414 1311111 88 5105100 95 10489 288 10777 2121111 89 5615610 96 10928 192 11120 1111211 90 6126120 97 11057 384 11441 3211111 91 6276270 11077 192 11269 11112101 92 6636630 98 11548 192 11740 1111121 93 7147140 99 11739 288 12027 2112111 94 7657650 100 12005 288 12293 1221111 95 8168160 101 12165 384 12549 5111111 96 8678670 102 12607 192 12799 1111112 97 9189180 103 12653 384 13037 2311111 98 9699690 104 19729 256 19985 11111111 99 11741730 105 20349 256 20605 111111101 100 13123110 106 20673 256 20929 111111011 101 14804790 107 21197 256 21453 1111111001 102 15825810 108 21457 256 21713 11111110001 103 16546530 109 21513 256 21769 1111110101 104 17687670 110 21772 256 22028 11111101001 105 18888870 111 22187 256 22443 111111100001 106 19399380 112 24905 384 25289 21111111 107 23483460 113 25621 384 26005 211111101 108 26246220 114 25986 384 26370 211111011 109 29099070 115 28540 384 28924 12111111 110* 35225190 116 29317 384 29701 121111101 111 38798760 117 31264 512 31776 31111111 112 46966920 118 32082 512 32594 311111101 113 48498450 119 33766 384 34150 11211111 114 58198140 120 35628 576 36204 22111111 115 67897830 121 37644 384 38028 11121111 116 77597520 122 39020 640 39660 41111111 117 87297210 123 40649 512 41161 13111111 118 96996900 124 41967 576 42543 21211111 119 106696590 125 43443 384 43827 11112111 120 116396280 126 44261 768 45029 32111111 121 126095970 127 45772 384 46156 11111211 122 135795660 128 46657 576 47233 21121111 123 145495350 129 47664 576 48240 12211111 124 155195040 130 48434 768 49202 51111111 125 164894730 131 49746 384 50130 11111121 126 174594420 132 50246 768 51014 23111111 127 184294110 133 51477 384 51861 11111112 128 193993800 134 51912 768 52680 31211111 129 203693490 135 52892 576 53468 12121111 130 213393180 136 53650 576 54226 21112111 131 223092870 137 82562 512 83074 111111111 132 281291010 138 85542 512 86054 1111111101 133 300690390 139 86466 512 86978 11111111001 134 340510170 140 87656 512 88168 1111111011 135 358888530 141 89086 512 89598 111111110001 136 397687290 142 90702 512 91214 1111111100001 137 417086670 143 91481 512 91993 11111111000001 138 446185740 144 102979 768 103747 211111111 139 562582020 145 106420 768 107188 2111111101 140 601380780 146 107499 768 108267 21111111001 141 669278610 147 117069 768 117837 121111111 142 843873030 148 120804 768 121572 1211111101 143 892371480 149 127820 1024 128844 311111111 144 1115464350 150 137221 768 137989 112111111 145 1338557220 151 144738 1152 145890 221111111 146 1561650090 152 152108 768 152876 111211111 147 1784742960 153 157880 1280 159160 411111111 148 2007835830 154 163870 1024 164894 131111111 149 2230928700 155 169035 1152 170187 212111111 150 2454021570 156 174326 768 175094 111121111 151 2677114440 157 178156 1536 179692 321111111 152 2900207310 158 183232 768 184000 111112111 153 3123300180 159 186933 1152 188085 211211111 154 3346393050 160 190797 1152 191949 122111111 155 3569485920 161 194091 1536 195627 511111111 156 3792578790 162 198382 768 199150 111111211 157 4015671660 163 200978 1536 202514 231111111 158 4238764530 164 204977 768 205745 111111121 159 4461857400 165 207323 1536 208859 312111111 160 4684950270 166 210701 1152 211853 121211111 161 4908043140 167 213587 1152 214739 211121111 162 5131136010 168 216766 768 217534 111111112 163 5354228880 169 218318 1920 220238 421111111 164 5577321750 170 221836 1024 222860 113111111 165 5800414620 171 224258 1152 225410 211112111 166 6023507490 172 226599 1280 227879 141111111 167 6246600360 173 228757 1536 230293 311211111 168 6469693230 174 348646 1024 349670 1111111111 169 6915878970 175 351974 1024 352998 11111111101 170 8254436190 176 361444 1024 362468 111111111001 171 8720021310 177 362536 1024 363560 11111111011 172 9146807670 178 367324 1024 368348 1111111110001 173 9592993410 179 370150 1024 371174 11111111100001 174 10407767370 180 372013 1024 373037 111111110101 175 10485364890 181 375602 1024 376626 111111111000001 176 11532931410 182 377920 1024 378944 1111111101001 177 11823922110 183 383315 1024 384339 1111111110000001 178 12929686770 184 384112 1024 385136 11111111001001 179 12939386460 185 430088 1536 431624 2111111111 180 13831757940 186 433944 1536 435480 21111111101 181 16508872380 187 444924 1536 446460 211111111001 182 17440042620 188 446047 1536 447583 21111111011 183 18293615340 189 451768 1536 453304 2111111110001 184 19185986820 190 455060 1536 456596 21111111100001 185 19409079690 191 485640 1536 487176 1211111111 186 20747636910 192 489829 1536 491365 12111111101 187 24763308570 193 501771 1536 503307 121111111001 188 25878772920 194 528344 2048 530392 3111111111 189 27663515880 195 532773 2048 534821 31111111101 190 32348466150 196 564703 1536 566239 1121111111 191 34579394850 197 569341 1536 570877 11211111101 192 38818159380 198 594834 2304 597138 2211111111 193 41495273820 199 599624 2304 601928 22111111101 194 45287852610 200 622881 1536 624417 1112111111 195 48411152790 201 627819 1536 629355 11121111101 196 51757545840 202 646388 2560 648948 4111111111 197 55327031760 203 651449 2560 654009 41111111101 198 58227239070 204 669249 2048 671297 1311111111 199 62242910730 205 674438 2048 676486 13111111101 200 64696932300 206 689574 2304 691878 2121111111 201 69158789700 207 694854 2304 697158 21211111101 202 71166625530 208 709456 1536 710992 1111211111 203 76074668670 209 714816 1536 716352 11112111101 204 77636318760 210 725761 3072 728833 3211111111 205 82990547640 211 731216 3072 734288 32111111101 206 84106011990 212 744066 1536 745602 1111121111 207 89906426610 213 749595 1536 751131 11111211101 208 90575705220 214 759134 2304 761438 2112111111 209 96822305580 215 764745 2304 767049 21121111101 210 97045398450 216 774141 2304 776445 1221111111 211 103515091680 217 787662 3072 790734 5111111111 212 109984784910 218 802832 1536 804368 1111112111 213 116454478140 219 814340 3072 817412 2311111111 214 122924171370 220 828394 1536 829930 1111111211 215 129393864600 221 838880 3072 841952 3121111111 216 135863557830 222 851229 2304 853533 1212111111 217 142333251060 223 862429 2304 864733 2111211111 218 148802944290 224 874008 1536 875544 1111111121 219 155272637520 225 882155 3840 885995 4211111111 220 161742330750 226 894086 2048 896134 1131111111 221 168212023980 227 903666 2304 905970 2111121111 222 174681717210 228 912961 2560 915521 1411111111 223 181151410440 229 921760 3072 924832 3112111111 224 187621103670 230 932340 1536 933876 1111111112 225 194090796900 231 939244 3456 942700 2221111111 226 200560490130 232 1454410 2048 1456458 11111111111 227 239378649510 233 1488865 2048 1490913 111111111101 228 255887521890 234 1500943 2048 1502991 111111111011 229 265257422430 235 1510476 2048 1512524 1111111111001 230 278196808890 236 1520871 2048 1522919 11111111110001 231 283551037770 237 1522564 2048 1524612 1111111110101 232 297382795710 238 1532974 2048 1535022 11111111101001 233 304075581810 239 1540983 2048 1543031 111111111100001 234 325046311590 240 1553131 2048 1555179 111111111010001 235 338431883790 241 1557242 2048 1559290 1111111110011 236 342893741190 242 1569498 2048 1571546 1111111111000001 237 366541585410 243 1581681 2048 1583729 1111111110100001 238 381711900570 244 1596245 2048 1598293 11111111110000001 239 394651287030 245 1604802 2048 1606850 111111111100000001 240 401120980260 246 1778346 3072 1781418 21111111111 241 478757299020 247 1818099 3072 1821171 211111111101 242 511775043780 248 1831946 3072 1835018 211111111011 243 530514844860 249 1843101 3072 1846173 2111111111001 244 556393617780 250 1855149 3072 1858221 21111111110001 245 567102075540 251 1856969 3072 1860041 2111111110101 246 594765591420 252 1869030 3072 1872102 21111111101001 247 601681470390 253 1997575 3072 2000647 12111111111 248 718135948530 254 2040712 3072 2043784 121111111101 249 767662565670 255 2055677 3072 2058749 121111111011 250 795772267290 256 2067874 3072 2070946 1211111111001 251 802241960520 257 2166613 4096 2170709 31111111111 252 957514598040 258 2212232 4096 2216328 311111111101 253 1002802450650 259 2308457 3072 2311529 11211111111 254 1196893247550 260 2356095 3072 2359167 112111111101 255 1203362940780 261 2428046 4608 2432654 22111111111 256 1403923430910 262 2536449 3072 2539521 11121111111 257 1604483921040 263 2630381 5120 2635501 41111111111 258 1805044411170 264 2718799 4096 2722895 13111111111 259 2005604901300 265 2798663 4608 2803271 21211111111 260 2206165391430 266 2874776 3072 2877848 11112111111 261 2406725881560 267 2941315 6144 2947459 32111111111 262 2607286371690 268 3009782 3072 3012854 11111211111 263 2807846861820 269 3069975 4608 3074583 21121111111 264 3008407351950 270 3128462 4608 3133070 12211111111 265 3208967842080 271 3182522 6144 3188666 51111111111 266 3409528332210 272 3238675 3072 3241747 11111121111 267 3610088822340 273 3286370 6144 3292514 23111111111 268 3810649312470 274 3338111 3072 3341183 11111112111 269 4011209802600 275 3381807 6144 3387951 31211111111 270 4211770292730 276 3428382 4608 3432990 12121111111 271 4412330782860 277 3471845 4608 3476453 21112111111 272 4612891272990 278 3515369 3072 3518441 11111111211 273 4813451763120 279 3551384 7680 3559064 42111111111 274 5014012253250 280 3594359 4096 3598455 11311111111 275 5214572743380 281 3632018 4608 3636626 21111211111 276 5415133233510 282 3668614 5120 3673734 14111111111 277 5615693723640 283 3703670 6144 3709814 31121111111 278 5816254213770 284 3741846 3072 3744918 11111111121 279 6016814703900 285 3772232 6912 3779144 22211111111 280 6217375194030 286 3809412 3072 3812484 11111111112 281 6417935684160 287 3837859 7168 3845027 61111111111 282 6618496174290 288 3872222 4608 3876830 12112111111 283 6819056664420 289 3903277 4608 3907885 21111121111 284 7019617154550 290 3933674 4608 3938282 11221111111 285 7220177644680 291 3959819 8192 3968011 33111111111 ////// Note ////// The sequence has been directly verified through term 110. All terms 1 - 285 were generated using the tier-3 algorithm, which uses a necessary but not sufficient condition that must be tested by A243822(m) = A010846(m) - A000005(m). ////// Observations ////// 1. Visually, in aggregate, the column MN(n) resembles MN(A244052). 2. Terms in n appear in A244052 except {78, 126, 990, 19110, 6276270, ...}. 3. Primorials A002110 seem to divide this sequence into tiers as they do A244052. 4. The function MN(A244052(n)) has been studied deeply enough to furnish A288784 and A288813 regarding candidate numbers for inclusion in A244052 (i.e., a necessary-but-insufficient condition for A244052). MN of the terms in n but not in A244052 involve turbulent terms t that exceed 2kp_T#, often with depth 1, except in the case of n = 78 which has depth 3. (See A244052). This seems to imply that a sequence similar to A288784 might be produced for this sequence that would serve a similar role that A288784 does for A244052. If this is true, then we might attain larger values of n without testing the full range of numbers up to a large limit. +-------------------------------------------------------------------+ | Table 2: Appearance of terms of A244052 among records of A243822. | +-------------------------------------------------------------------+ k = index in this table. m = A244052(k). i = index of A293555. s = A243822(m). tau(m) = A000005(m). r = A010846(m) = A243822(m) + A000005(m). MN(n) = multiplicity notation = rev(A054841(n)), shorthand notation that lists exponents e of p, pi(p) conveyed by position from left. e.g., 60 = 2^2 * 3 * 5 = "211". k m i s(m) tau(m) r(m) MN(m) --------------------------------------------------------------------- 1 1 1 0 1 1 0 2 2 0 2 2 1 3 4 0 3 3 2 4 6 2 1 4 5 11 5 10 3 2 4 6 101 6 12 2 6 8 21 7 18 4 4 6 10 12 8 24 3 8 11 31 9 30 5 10 8 18 111 10 42 6 11 8 19 1101 11 60 7 14 12 26 211 12 84 9 16 12 28 2101 13 90 10 20 12 32 121 14 120 20 16 36 311 15 150 12 29 12 41 112 16 180 26 18 44 221 17 210 13 52 16 68 1111 18 330 14 61 16 77 11101 19 390 15 64 16 80 111001 20 420 16 72 24 96 2111 21 630 17 91 24 115 1211 22 840 18 99 32 131 3111 23 1050 20 121 24 145 1121 24 1260 120 36 156 2211 25 1470 21 142 24 166 1112 26 1680 134 40 174 4111 27 1890 22 151 32 183 1311 28 2100 23 156 36 192 2121 29 2310 24 251 32 283 11111 30 2730 25 263 32 295 111101 31 3570 26 281 32 313 1111001 32 3990 27 290 32 322 11110001 33 4620 28 334 48 382 21111 34 5460 29 347 48 395 211101 35 6930 30 404 48 452 12111 36 8190 31 415 48 463 121101 37 9240 32 441 64 505 31111 38 10920 33 455 64 519 311101 39 11550 34 503 48 551 11211 40 13650 35 519 48 567 112101 41 13860 36 521 72 593 22111 42 16170 37 581 48 629 11121 43 18480 580 80 660 41111 44 20790 39 627 64 691 13111 45 23100 40 645 72 717 21211 46 25410 41 695 48 743 11112 47 27720 670 96 766 32111 48 30030 42 1097 64 1161 111111 49 39270 43 1160 64 1224 1111101 50 43890 44 1189 64 1253 11111001 51 46410 45 1193 64 1257 1111011 52 51870 46 1221 64 1285 11110101 53 53130 47 1242 64 1306 111110001 54 60060 48 1430 96 1526 211111 55 78540 49 1501 96 1597 2111101 56 87780 50 1535 96 1631 21111001 57 90090 51 1683 96 1779 121111 58 117810 52 1760 96 1856 1211101 59 120120 53 1849 128 1977 311111 60 150150 54 2048 96 2144 112111 61 180180 55 2150 144 2294 221111 62 210210 56 2324 96 2420 111211 63 240240 57 2378 160 2538 411111 64 270270 58 2517 128 2645 131111 65 300300 59 2599 144 2743 212111 66 330330 60 2740 96 2836 111121 67 360360 2729 192 2921 321111 68 390390 61 2905 96 3001 111112 69 420420 62 2936 144 3080 211211 70 450450 63 3009 144 3153 122111 71 480480 64 3031 192 3223 511111 72 510510 65 4715 128 4843 1111111 73 570570 66 4811 128 4939 11111101 74 690690 67 4991 128 5119 111111001 75 746130 68 5010 128 5138 11111011 76 870870 69 5236 128 5364 1111110001 77 930930 70 5308 128 5436 11111100001 78 1021020 71 6033 192 6225 2111111 79 1141140 72 6145 192 6337 21111101 80 1381380 73 6354 192 6546 211111001 81 1492260 74 6368 192 6560 21111011 82 1531530 75 6986 192 7178 1211111 83 1711710 76 7107 192 7299 12111101 84 2042040 77 7672 256 7928 3111111 85 2282280 78 7799 256 8055 31111101 86 2552550 79 8361 192 8553 1121111 87 2852850 80 8493 192 8685 11211101 88 3063060 81 8811 288 9099 2211111 89 3423420 82 8948 288 9236 22111101 90 3573570 83 9388 192 9580 1112111 91 3993990 84 9527 192 9719 11121101 92 4084080 85 9690 320 10010 4111111 93 4564560 86 9835 320 10155 41111101 94 4594590 87 10158 256 10414 1311111 95 5105100 88 10489 288 10777 2121111 96 5615610 89 10928 192 11120 1111211 97 6126120 90 11057 384 11441 3211111 98 6636630 92 11548 192 11740 1111121 99 7147140 93 11739 288 12027 2112111 100 7657650 94 12005 288 12293 1221111 101 8168160 95 12165 384 12549 5111111 102 8678670 96 12607 192 12799 1111112 103 9189180 97 12653 384 13037 2311111 104 9699690 98 19729 256 19985 11111111 105 11741730 99 20349 256 20605 111111101 106 13123110 100 20673 256 20929 111111011 107 14804790 101 21197 256 21453 1111111001 108 15825810 102 21457 256 21713 11111110001 109 16546530 103 21513 256 21769 1111110101 110 17687670 104 21772 256 22028 11111101001 111 18888870 105 22187 256 22443 111111100001 112 19399380 106 24905 384 25289 21111111 113 23483460 107 25621 384 26005 211111101 114 26246220 108 25986 384 26370 211111011 115 29099070 109 28540 384 28924 12111111 116* 35225190 110 29317 384 29701 121111101 117 38798760 111 31264 512 31776 31111111 118 46966920 112 32082 512 32594 311111101 119 48498450 113 33766 384 34150 11211111 120 58198140 114 35628 576 36204 22111111 121 67897830 115 37644 384 38028 11121111 122 77597520 116 39020 640 39660 41111111 123 87297210 117 40649 512 41161 13111111 124 96996900 118 41967 576 42543 21211111 125 106696590 119 43443 384 43827 11112111 126 116396280 120 44261 768 45029 32111111 127 126095970 121 45772 384 46156 11111211 128 135795660 122 46657 576 47233 21121111 129 145495350 123 47664 576 48240 12211111 130 155195040 124 48434 768 49202 51111111 131 164894730 125 49746 384 50130 11111121 132 174594420 126 50246 768 51014 23111111 133 184294110 127 51477 384 51861 11111112 134 193993800 128 51912 768 52680 31211111 135 203693490 129 52892 576 53468 12121111 136 213393180 130 53650 576 54226 21112111 137 223092870 131 82562 512 83074 111111111 138 281291010 132 85542 512 86054 1111111101 139 300690390 133 86466 512 86978 11111111001 140 340510170 134 87656 512 88168 1111111011 141 358888530 135 89086 512 89598 111111110001 142 397687290 136 90702 512 91214 1111111100001 143 417086670 137 91481 512 91993 11111111000001 144 446185740 138 102979 768 103747 211111111 145 562582020 139 106420 768 107188 2111111101 146 601380780 140 107499 768 108267 21111111001 147 669278610 141 117069 768 117837 121111111 148 843873030 142 120804 768 121572 1211111101 149 892371480 143 127820 1024 128844 311111111 150 1115464350 144 137221 768 137989 112111111 151 1338557220 145 144738 1152 145890 221111111 152 1561650090 146 152108 768 152876 111211111 153 1784742960 147 157880 1280 159160 411111111 154 2007835830 148 163870 1024 164894 131111111 155 2230928700 149 169035 1152 170187 212111111 156 2454021570 150 174326 768 175094 111121111 157 2677114440 151 178156 1536 179692 321111111 158 2900207310 152 183232 768 184000 111112111 159 3123300180 153 186933 1152 188085 211211111 160 3346393050 154 190797 1152 191949 122111111 161 3569485920 155 194091 1536 195627 511111111 162 3792578790 156 198382 768 199150 111111211 163 4015671660 157 200978 1536 202514 231111111 164 4238764530 158 204977 768 205745 111111121 165 4461857400 159 207323 1536 208859 312111111 166 4684950270 160 210701 1152 211853 121211111 167 4908043140 161 213587 1152 214739 211121111 168 5131136010 162 216766 768 217534 111111112 169 5354228880 163 218318 1920 220238 421111111 170 5577321750 164 221836 1024 222860 113111111 171 5800414620 165 224258 1152 225410 211112111 172 6023507490 166 226599 1280 227879 141111111 173 6246600360 167 228757 1536 230293 311211111 174 6469693230 168 348646 1024 349670 1111111111 175 6915878970 169 351974 1024 352998 11111111101 176 8254436190 170 361444 1024 362468 111111111001 177 8720021310 171 362536 1024 363560 11111111011 178 9146807670 172 367324 1024 368348 1111111110001 179 9592993410 173 370150 1024 371174 11111111100001 180 10407767370 174 372013 1024 373037 111111110101 181 10485364890 175 375602 1024 376626 111111111000001 182 11532931410 176 377920 1024 378944 1111111101001 183 11823922110 177 383315 1024 384339 1111111110000001 184 12929686770 178 384112 1024 385136 11111111001001 185 12939386460 179 430088 1536 431624 2111111111 186 13831757940 180 433944 1536 435480 21111111101 187 16508872380 181 444924 1536 446460 211111111001 188 17440042620 182 446047 1536 447583 21111111011 189 18293615340 183 451768 1536 453304 2111111110001 190 19185986820 184 455060 1536 456596 21111111100001 191 19409079690 185 485640 1536 487176 1211111111 192 20747636910 186 489829 1536 491365 12111111101 193 24763308570 187 501771 1536 503307 121111111001 194 25878772920 188 528344 2048 530392 3111111111 195 27663515880 189 532773 2048 534821 31111111101 196 32348466150 190 564703 1536 566239 1121111111 197 34579394850 191 569341 1536 570877 11211111101 198 38818159380 192 594834 2304 597138 2211111111 199 41495273820 193 599624 2304 601928 22111111101 200 45287852610 194 622881 1536 624417 1112111111 201 48411152790 195 627819 1536 629355 11121111101 202 51757545840 196 646388 2560 648948 4111111111 203 55327031760 197 651449 2560 654009 41111111101 204 58227239070 198 669249 2048 671297 1311111111 205 62242910730 199 674438 2048 676486 13111111101 206 64696932300 200 689574 2304 691878 2121111111 207 69158789700 201 694854 2304 697158 21211111101 208 71166625530 202 709456 1536 710992 1111211111 209 76074668670 203 714816 1536 716352 11112111101 210 77636318760 204 725761 3072 728833 3211111111 211 82990547640 205 731216 3072 734288 32111111101 212 84106011990 206 744066 1536 745602 1111121111 213 89906426610 207 749595 1536 751131 11111211101 214 90575705220 208 759134 2304 761438 2112111111 215 96822305580 209 764745 2304 767049 21121111101 216 97045398450 210 774141 2304 776445 1221111111 217 103515091680 211 787662 3072 790734 5111111111 218 109984784910 212 802832 1536 804368 1111112111 219 116454478140 213 814340 3072 817412 2311111111 220 122924171370 214 828394 1536 829930 1111111211 221 129393864600 215 838880 3072 841952 3121111111 222 135863557830 216 851229 2304 853533 1212111111 223 142333251060 217 862429 2304 864733 2111211111 224 148802944290 218 874008 1536 875544 1111111121 225 155272637520 219 882155 3840 885995 4211111111 226 161742330750 220 894086 2048 896134 1131111111 227 168212023980 221 903666 2304 905970 2111121111 228 174681717210 222 912961 2560 915521 1411111111 229 181151410440 223 921760 3072 924832 3112111111 230 187621103670 224 932340 1536 933876 1111111112 231 194090796900 225 939244 3456 942700 2221111111 232 200560490130 226 1454410 2048 1456458 11111111111 233 239378649510 227 1488865 2048 1490913 111111111101 234 255887521890 228 1500943 2048 1502991 111111111011 235 265257422430 229 1510476 2048 1512524 1111111111001 236 278196808890 230 1520871 2048 1522919 11111111110001 237 283551037770 231 1522564 2048 1524612 1111111110101 238 297382795710 232 1532974 2048 1535022 11111111101001 239 304075581810 233 1540983 2048 1543031 111111111100001 240 325046311590 234 1553131 2048 1555179 111111111010001 241 338431883790 235 1557242 2048 1559290 1111111110011 242 342893741190 236 1569498 2048 1571546 1111111111000001 243 366541585410 237 1581681 2048 1583729 1111111110100001 244 381711900570 238 1596245 2048 1598293 11111111110000001 245 394651287030 239 1604802 2048 1606850 111111111100000001 246 401120980260 240 1778346 3072 1781418 21111111111 247 478757299020 241 1818099 3072 1821171 211111111101 248 511775043780 242 1831946 3072 1835018 211111111011 249 530514844860 243 1843101 3072 1846173 2111111111001 250 556393617780 244 1855149 3072 1858221 21111111110001 251 567102075540 245 1856969 3072 1860041 2111111110101 252 594765591420 246 1869030 3072 1872102 21111111101001 253 601681470390 247 1997575 3072 2000647 12111111111 254 718135948530 248 2040712 3072 2043784 121111111101 255 767662565670 249 2055677 3072 2058749 121111111011 256 795772267290 250 2067874 3072 2070946 1211111111001 257 802241960520 251 2166613 4096 2170709 31111111111 258 957514598040 252 2212232 4096 2216328 311111111101 259 1002802450650 253 2308457 3072 2311529 11211111111 260 1196893247550 254 2356095 3072 2359167 112111111101 261 1203362940780 255 2428046 4608 2432654 22111111111 262 1403923430910 256 2536449 3072 2539521 11121111111 263 1604483921040 257 2630381 5120 2635501 41111111111 264 1805044411170 258 2718799 4096 2722895 13111111111 265 2005604901300 259 2798663 4608 2803271 21211111111 266 2206165391430 260 2874776 3072 2877848 11112111111 267 2406725881560 261 2941315 6144 2947459 32111111111 268 2607286371690 262 3009782 3072 3012854 11111211111 269 2807846861820 263 3069975 4608 3074583 21121111111 270 3008407351950 264 3128462 4608 3133070 12211111111 271 3208967842080 265 3182522 6144 3188666 51111111111 272 3409528332210 266 3238675 3072 3241747 11111121111 273 3610088822340 267 3286370 6144 3292514 23111111111 274 3810649312470 268 3338111 3072 3341183 11111112111 275 4011209802600 269 3381807 6144 3387951 31211111111 276 4211770292730 270 3428382 4608 3432990 12121111111 277 4412330782860 271 3471845 4608 3476453 21112111111 278 4612891272990 272 3515369 3072 3518441 11111111211 279 4813451763120 273 3551384 7680 3559064 42111111111 280 5014012253250 274 3594359 4096 3598455 11311111111 281 5214572743380 275 3632018 4608 3636626 21111211111 282 5415133233510 276 3668614 5120 3673734 14111111111 283 5615693723640 277 3703670 6144 3709814 31121111111 284 5816254213770 278 3741846 3072 3744918 11111111121 285 6016814703900 279 3772232 6912 3779144 22211111111 286 6217375194030 280 3809412 3072 3812484 11111111112 287 6417935684160 281 3837859 7168 3845027 61111111111 288 6618496174290 282 3872222 4608 3876830 12112111111 289 6819056664420 283 3903277 4608 3907885 21111121111 290 7019617154550 284 3933674 4608 3938282 11221111111 291 7220177644680 285 3959819 8192 3968011 33111111111 ////// Note ////// The sequence has been directly verified through term 116. All terms 1 - 291 were generated using the tier-3 algorithm, which uses a necessary but not sufficient condition that must be tested by A243822(m) = A010846(m) - A000005(m). ////// Observations ////// 1. Terms m in A244052 appear in n except {2, 4, 12, 24, 120, 180, 1260, 1680, 18480, 27720, 360360, ...}. 2. Terms m in A244052 not in n seem to have tau(m) significantly greater than surrounding values. ////// Revision Record ////// 201710101400 created, range 1 <= n <= 10^6. 201710110700 range increased to 1 <= n <= 3 * 10^6. 201710111500 range increased to 1 <= n <= 5 * 10^6. 201710111915 A293555 and A293556 reserved. 201710120545 range increased to 1 <= n <= 7 * 10^6. 201710121200 range increased to 1 <= n <= 8 * 10^6. 201710130230 range increased to 1 <= n <= 9 * 10^6. 201710140130 range increased to 1 <= n <= 10^7. 201710151930 range increased to 1 <= n <= 15 * 10^6. 201710160730 range increased to 1 <= n <= 17 * 10^6. 201710161545 range increased to 1 <= n <= 18 * 10^6. 201710171015 range increased to 1 <= n <= 21 * 10^6. ----- 201710172100 tier-3 algorithm used up to A002110(10) - 1. 201710180800 tier-3 algorithm used up to A002110(12) - 1. 201710241130 verified to 26 * 10^6. 201710261300 verified to 29 * 10^6. 201710271845 verified to 32 * 10^6. 201711070330 verified to 36 * 10^6. 201711171130 finalize revisions. (eof)