Index k of terms m in A330781 that are also in the Chernoff Sequence (A006939). Michael Thomas De Vlieger, St. Louis, Missouri, 202001161230. Concerns sequences A006939, A067255, A322156, A330781. Numbers m in A006939 have a prime signature that monotonically decreases. Using the "multiplicity notation" of row m in A067255, we can write m as follows: A006939(1) = 1 => *empty product* => 0 A006939(2) = 2 => 2^1 => 1 A006939(3) = 12 => 2^2 * 3^1 => 2.1 A006939(4) = 360 => 2^3 * 3^2 * 5^1 => 3.2.1 A006939(5) = 75600 => 2^4 * 3^3 * 5^2 * 7^1 => 4.3.2.1 etc. We can abbreviate terms in A330781 using "Durfee formulation" notation described in A322156. For instance, given m = 360 => 3.2.1, we can chart its prime signature as 3 x 2 X X 1 X X x 2 3 5 and thus use a row j in A322156 to encode it. Here we place a 2-square (shown by X's) at origin, then two 1-squares (shown by x's) above and to the left, thus abbreviating 360 as 2.1. A006939(1) = 1 => 0 => 0 A006939(2) = 2 => 1 => 1 A006939(3) = 12 => 2.1 => 1.1 A006939(4) = 360 => 3.2.1 => 2.1 A006939(5) = 75600 => 4.3.2.1 => 2.1.1 etc. The algorithm that yields A322156 is described at the end of this document, and is available at https://oeis.org/A322156. In this document we use the following registers: n index in this document. j row in A322156 that contains the terms in column A322156(m). k index of m = A006939(n) in A330781. * We list only the first few terms of A006939, since decimally, these quickly become cumbersome. n j A322156(m) k A006939(n)* --------------------------------------------------------------- 1 1 1 1 1 2 2 1.1 2 2 3 4 2.1 4 12 4 5 2.1.1 6 360 5 9 3.1.1 9 75600 6 11 3.2.1 13 174636000 7 17 4.2.1 19 5244319080000 8 18 4.2.1.1 23 ... 9 28 5.2.1.1 32 10 32 5.3.1.1 40 11 46 6.3.1.1 50 12 48 6.3.2.1 64 13 68 7.3.2.1 81 14 74 7.4.2.1 97 15 100 8.4.2.1 117 16 101 8.4.2.1.1 140 17 137 9.4.2.1.1 167 18 147 9.5.2.1.1 197 19 193 10.5.2.1.1 233 20 197 10.5.3.1.1 270 21 257 11.5.3.1.1 316 22 271 11.6.3.1.1 366 23 345 12.6.3.1.1 425 24 347 12.6.3.2.1 484 25 441 13.6.3.2.1 559 26 461 13.7.3.2.1 639 27 575 14.7.3.2.1 726 28 581 14.7.4.2.1 822 29 721 15.7.4.2.1 926 30 747 15.8.4.2.1 1041 31 913 16.8.4.2.1 1168 32 914 16.8.4.2.1.1 1303 33 1116 17.8.4.2.1.1 1452 34 1152 17.9.4.2.1.1 1618 35 1390 18.9.4.2.1.1 1802 36 1400 18.9.5.2.1.1 1994 37 1684 19.9.5.2.1.1 2209 38 1730 19.10.5.2.1.1 2442 39 2060 20.10.5.2.1.1 2690 40 2064 20.10.5.3.1.1 2959 41 2454 21.10.5.3.1.1 3254 42 2514 21.11.5.3.1.1 3574 43 2964 22.11.5.3.1.1 3922 44 2978 22.11.6.3.1.1 4288 45 3502 23.11.6.3.1.1 4685 46 3576 23.12.6.3.1.1 5102 47 4174 24.12.6.3.1.1 5562 48 4176 24.12.6.3.2.1 6044 49 4868 25.12.6.3.2.1 6584 50 4962 25.13.6.3.2.1 7133 51 5748 26.13.6.3.2.1 7740 52 5768 26.13.7.3.2.1 8383 53 6668 27.13.7.3.2.1 9060 54 6782 27.14.7.3.2.1 9783 55 7796 28.14.7.3.2.1 10570 56 7802 28.14.7.4.2.1 11386 57 8956 29.14.7.4.2.1 12263 58 9096 29.15.7.4.2.1 13185 59 10390 30.15.7.4.2.1 14177 60 10416 30.15.8.4.2.1 15208 61 11876 31.15.8.4.2.1 16319 62 12042 31.16.8.4.2.1 17497 63 13668 32.16.8.4.2.1 18732 64 13669 32.16.8.4.2.1.1 20044 65 15497 33.16.8.4.2.1.1 21435 66 15699 33.17.8.4.2.1.1 22880 67 17729 34.17.8.4.2.1.1 24446 68 17765 34.17.9.4.2.1.1 26079 69 20033 35.17.9.4.2.1.1 27813 70 20271 35.18.9.4.2.1.1 29610 71 22777 36.18.9.4.2.1.1 31540 72 22787 36.18.9.5.2.1.1 33557 73 25577 37.18.9.5.2.1.1 35684 74 25861 37.19.9.5.2.1.1 37931 75 28935 38.19.9.5.2.1.1 40255 76 28981 38.19.10.5.2.1.1 42728 77 32385 39.19.10.5.2.1.1 45325 78 32715 39.20.10.5.2.1.1 48065 79 36449 40.20.10.5.2.1.1 50929 80 36453 40.20.10.5.3.1.1 53910 81 40577 41.20.10.5.3.1.1 57058 82 40967 41.21.10.5.3.1.1 60353 83 45481 42.21.10.5.3.1.1 63829 84 45541 42.21.11.5.3.1.1 67447 85 50505 43.21.11.5.3.1.1 71248 86 50955 43.22.11.5.3.1.1 75220 87 56369 44.22.11.5.3.1.1 79381 88 56383 44.22.11.6.3.1.1 83724 89 62321 45.22.11.6.3.1.1 88250 90 62845 45.23.11.6.3.1.1 92997 91 69307 46.23.11.6.3.1.1 97964 92 69381 46.23.12.6.3.1.1 103143 93 76441 47.23.12.6.3.1.1 108552 94 77039 47.24.12.6.3.1.1 114194 95 84697 48.24.12.6.3.1.1 120072 96 84699 48.24.12.6.3.2.1 126225 97 93049 49.24.12.6.3.2.1 132639 98 93741 49.25.12.6.3.2.1 139311 99 102783 50.25.12.6.3.2.1 146231 100 102877 50.25.13.6.3.2.1 153501 101 112705 51.25.13.6.3.2.1 161050 102 113491 51.26.13.6.3.2.1 168897 103 124105 52.26.13.6.3.2.1 177086 104 124125 52.26.13.7.3.2.1 185555 105 135639 53.26.13.7.3.2.1 194435 106 136539 53.27.13.7.3.2.1 203645 107 148953 54.27.13.7.3.2.1 213212 108 149067 54.27.14.7.3.2.1 223121 109 162495 55.27.14.7.3.2.1 233450 110 163509 55.28.14.7.3.2.1 244178 111 177951 56.28.14.7.3.2.1 255304 112 177957 56.28.14.7.4.2.1 266833 113 193553 57.28.14.7.4.2.1 278848 114 194707 57.29.14.7.4.2.1 291341 115 211457 58.29.14.7.4.2.1 304289 116 211597 58.29.15.7.4.2.1 317640 117 229641 59.29.15.7.4.2.1 331570 118 230935 59.30.15.7.4.2.1 345887 119 250273 60.30.15.7.4.2.1 360842 120 250299 60.30.15.8.4.2.1 376291 121 271097 61.30.15.8.4.2.1 392327 122 272557 61.31.15.8.4.2.1 408934 123 294815 62.31.15.8.4.2.1 426093 124 294981 62.31.16.8.4.2.1 443911 125 318865 63.31.16.8.4.2.1 462260 126 320491 63.32.16.8.4.2.1 481310 127 346001 64.32.16.8.4.2.1 501022 128 346002 64.32.16.8.4.2.1.1 521409 129 373340 65.32.16.8.4.2.1.1 542474 130 375168 65.33.16.8.4.2.1.1 564276 131 404334 66.33.16.8.4.2.1.1 586879 132 404536 66.33.17.8.4.2.1.1 610144 133 435732 67.33.17.8.4.2.1.1 634293 134 437762 67.34.17.8.4.2.1.1 659140 135 470988 68.34.17.8.4.2.1.1 684914 136 471024 68.34.17.9.4.2.1.1 711422 137 506518 69.34.17.9.4.2.1.1 738814 138 508786 69.35.17.9.4.2.1.1 767189 139 546548 70.35.17.9.4.2.1.1 796389 140 546786 70.35.18.9.4.2.1.1 826591 141 587054 71.35.18.9.4.2.1.1 857701 142 589560 71.36.18.9.4.2.1.1 889837 143 632334 72.36.18.9.4.2.1.1 923028 ===== A322156 Algorithm ===== Let S be a sequence starting with n. Let k be the index of a term in S, with n at position k = 0. Let S_r be the r-th sequence in row n. Starting with S_1 = {n}, we either (A) append a 1 to the left of S_r, or (B) we drop the most recently-appended term S_(k) and increment the rightmost term (k - 1). By default we execute (A) and test according to the following. Consider the reversed accumulation A_(r + 1) = Sum(reverse(S_(k + 1))) = Sum(k_m, k_(m - 1), ..., k_2, k_1). If S_r - A_(r + 1) contains nothing less than 0, then S_(k + 1) is retained, else we execute (B). We end after k_1 = n, since otherwise we would enter an endless loop that also increments k_0 ad infinitum. The first sequence S in row n is {n} while the last is {n, n}. All rows n contain {{n}, {n, 1}, {n, n}}. Only one repeated term k may appear at the end of any S in row n. The longest possible sequence S in row n has 2 + floor(log2(n)) terms = 2 + A113473(n). The sequence S describes unique integer partitions L that are recursively symmetrical. Example: We can convert S = {4, 2, 1} into the partition (7, 6, 5, 4, 3, 2, 1), a partition of N = 28. We set a 4X Durfee square with its upper-left corner at origin. Then we set 2^k = 2^1 = 2 2X squares with its upper-left corner in any coordinate bounded at left and top by either a previously-lain square or an axis. Finally, we set 2^2 = 4 1X squares as above once again. We obtain a Ferrer diagram as below, with the k marked, i.e., the 1st term 4X, the 2nd term 2X, the 3rd term 1X squares: 0 0 0 0 1 1 2 0 0 0 0 1 1 0 0 0 0 2 0 0 0 0 1 1 2 1 1 2 The resulting partition L is recursively self-conjugate; its arms are identical to its legs. We can eliminate the Durfee square and the other appendage and have a symmetrical partition L_1 with Durfee square of k_1 units, etc. Were we to admit either more than 1 repeated k or a term such that S_k - A_(k + 1) had differences less than 1, we would have overlapping squares in the Ferrer diagram. Such diagrams are generated by larger n and all resulting diagrams are unique given the described algorithm. (eof)