+--------------------------------------------------------------------------+ | Number of terms of A288813 that have j = m - pi(A053669(A288813(m))) + 1 | +--------------------------------------------------------------------------+ Written by Michael Thomas De Vlieger, St. Louis, Missouri, USA, 20170626. a(n) = A288813(n) = Irregular triangle read by rows: T(m,k) is the list of numbers A002110(m) <= t < 2*A002110(m) such that A001221(t) = m. a(n) produces the terms k*t with k = 1 in A288784. In this work, m is called a "tier" or row of irregular triangle T(m,k). All terms of a(n) are deemed "level 1" and "primary turbulent" according to the document https://oeis.org/A288784/a288784.txt. Let the numbers a(n) = t = {3, 10, 42, 330, 390, 2730, 3570, 3990, 4290, ...} be "turbulent" numbers on account of the aggregate appearance of A054841(a(n)) in A288784. These numbers are "turbulent" because pi(A006530(a(n))) > m. +-------+ | DATA. | +-------+ "n" is the position in a(n) = n, (position of A288813(n) in A288813). "m" is the index of the largest primorial p_m# = A002110(m) < a(n). All terms of p_m# <= a(n) < p_(m+1)# must have A001221(a(n)) = m. "j": "Depth" j = n - pi(A053669(A288784(x))) + 1. This is the position of first "0" in A054841(a(n)) minus m plus 1. The variable j is descriptive of the greatest common divisor A002110(j - 1) of all terms of T(m,k) for a given tier m. 1| 1 2| 1 3| 1 4| 2 5| 3 1 6| 3 2 7| 4 3 1 8| 4 3 1 9| 5 3 2 10| 6 7 2 1 11| 7 9 3 1 12| 9 11 8 2 13| 9 16 11 6 1 14| 9 16 15 6 3 15| 9 15 14 9 3 1 16| 11 20 16 13 6 2 17| 13 28 24 16 12 6 1 18| 12 33 32 18 10 7 3 19| 13 36 48 33 13 7 4 2 20| 14 40 50 49 27 10 5 3 1 21| 13 38 54 43 34 15 5 2 1 22| 15 40 57 56 32 24 9 3 1 23| 15 43 56 59 44 21 15 5 1 24| 16 49 70 58 52 35 14 10 3 25| 19 64 91 96 66 54 33 15 12 2 26| 20 78 130 124 107 61 50 29 14 10 1 27| 19 76 150 153 117 89 44 33 18 10 6 28| 19 72 140 186 137 89 63 29 23 10 5 3 29| 18 65 113 153 149 86 48 31 15 10 4 1 1 m 30| 18 59 95 106 114 93 46 22 12 4 4 2 31| 23 78 123 141 129 124 93 40 20 11 5 4 1 32| 23 113 162 165 162 130 116 82 34 17 9 3 2 33| 25 118 264 220 190 170 126 108 73 28 14 6 1 1 34| 25 121 258 363 204 158 132 89 74 48 17 6 3 35| 27 131 310 431 471 213 156 126 85 72 45 14 5 2 36| 26 148 314 485 495 463 169 118 94 62 50 30 8 2 1 37| 28 148 383 491 590 506 420 134 92 73 45 37 21 4 1 38| 28 162 390 665 614 643 485 380 110 73 58 36 27 16 3 39| 28 169 430 639 839 623 590 409 309 79 51 43 23 17 9 1 40| 28 175 472 745 829 923 583 524 352 254 58 39 30 16 13 5 41| 30 187 501 863 1016 940 947 535 469 302 211 48 29 24 12 11 4 42| 30 187 514 841 1052 1013 818 769 395 337 205 143 29 18 15 6 4 2 43| 32 197 585 1011 1223 1301 1103 818 747 364 309 184 126 23 14 11 6 5 1 44| 32 213 581 1069 1343 1324 1244 950 665 589 275 227 129 88 13 9 5 4 2 45| 32 205 630 1002 1375 1394 1182 1013 728 475 419 180 148 79 55 7 4 3 1 1 46| 32 191 544 1010 1101 1236 1052 804 641 433 267 228 90 73 37 23 2 1 1 47| 35 210 580 1013 1442 1239 1264 978 691 540 352 210 177 70 55 26 14 1 48| 38 274 746 1339 1801 2178 1642 1589 1189 821 627 399 241 202 80 60 29 18 49| 38 310 1031 1683 2319 2599 2847 1939 1815 1304 877 664 418 255 211 77 61 28 19 50| 38 301 1154 2282 2581 2992 2945 2982 1871 1694 1183 773 575 362 211 179 60 45 19 12 51| 39 287 1081 2589 3508 3054 3180 2868 2766 1607 1429 966 620 451 273 155 126 40 30 11 6 52| 39 298 1035 2447 4234 4394 3197 3123 2670 2479 1369 1202 796 494 355 212 111 91 25 20 6 3 53| 39 290 990 2127 3650 4972 4175 2632 2455 2000 1799 953 817 529 317 220 126 60 52 11 8 2 1 54| 41 305 1062 2248 3553 4976 5795 4228 2445 2228 1774 1581 807 690 435 261 176 98 45 36 8 5 1 55| 42 340 1131 2484 3801 4895 5982 6250 4111 2241 2014 1583 1391 687 588 371 216 145 77 34 29 6 3 56| 43 357 1320 2653 4343 5360 5980 6641 6427 3927 2047 1827 1413 1233 598 517 315 181 119 61 29 22 4 2 57| 42 373 1417 3243 4642 6270 6647 6748 7007 6419 3717 1872 1663 1276 1110 529 455 276 156 101 52 25 18 2 1 58| 42 368 1409 3266 5333 5920 6984 6571 6194 6119 5362 2942 1424 1251 949 834 386 326 193 103 69 34 14 11 1 59| 42 361 1405 3266 5415 7012 6501 7001 6070 5429 5174 4379 2307 1075 940 707 613 277 230 134 71 45 22 8 7 60| 42 367 1321 3141 5204 6792 7399 6000 6053 4930 4237 3936 3236 1639 733 635 471 404 178 148 81 43 26 12 4 3 ------------------------------------------------------------------------------------------------------------------------------------------- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 "depth" j