Index k of terms m in A330781 that are also in A181555 = A002110(n)^n. Michael Thomas De Vlieger, St. Louis, Missouri, 202001161430. Concerns sequences A000123, A002110, A067255, A131205, A181555, A322156, A330781. Numbers m in A181555 have a prime signature that monotonically decreases. Using the "multiplicity notation" of row m in A067255, we can write m as follows: A181555(1) = 1 => *empty product* => 0 A181555(2) = 2 => 2^1 => 1 A181555(3) = 36 => 2^2 * 3^2 => 2.2 A181555(4) = 27000 => 2^3 * 3^3 * 5^3 => 3.3.3 A181555(5) = 1944810000 => 2^4 * 3^4 * 5^4 * 7^4 => 4.4.4.4 etc. We can abbreviate terms in A330781 using "Durfee formulation" notation described in A322156. For instance, given m = 27000 => 3.3.3, we can chart its prime signature as 3 X X X 2 X X X 1 X X X 2 3 5 and thus use a row j in A322156 to encode it. In the case of m in A181555, the number can be represented by a square with side length D. In A322156, these are rows with a single term, which appear at j in A131205, the partial sums of A000123: Number of binary partitions: number of partitions of 2n into powers of 2. A181555(1) = 1 => 0 => 0 A181555(2) = 2 => 1 => 1 A181555(3) = 36 => 2.2 => 2 A181555(4) = 27000 => 3.3.3 => 3 A181555(5) = 1944810000 => 4.4.4.4 => 4 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 index of m = A181555(n) in A330781. k row in A322156 that contains only n as a term = A131205(n). * We list only the first few terms of A181555, since decimally, these quickly become cumbersome. n j k A181555(n)* -------------------------------------------- 0 1 0 1 1 2 1 2 2 4 3 36 3 6 7 27000 4 11 13 1944810000 5 17 23 65774855015100000 6 26 37 ... 7 38 57 8 52 83 9 72 119 10 97 165 11 128 225 12 163 299 13 211 393 14 262 507 15 328 647 16 409 813 17 492 1015 18 603 1253 19 728 1537 20 861 1867 21 1024 2257 22 1214 2707 23 1421 3231 24 1657 3829 25 1927 4521 26 2233 5307 27 2580 6207 28 2951 7221 29 3382 8375 30 3863 9669 31 4389 11129 32 4996 12755 33 5633 14583 34 6364 16613 35 7160 18881 36 8035 21387 37 8998 24177 38 10046 27251 39 11208 30655 40 12472 34389 41 13825 38513 42 15329 43027 43 16953 47991 44 18719 53405 45 20632 59343 46 22673 65805 47 24927 72865 48 27332 80523 49 29954 88873 50 32762 97915 51 35776 107743 52 39052 118357 53 42525 129871 54 46292 142285 55 50290 155713 56 54604 170155 57 59258 185751 58 64144 202501 59 69435 220545 60 75048 239883 61 81028 260681 62 87458 282939 63 94266 306823 64 101579 332333 65 109281 359671 66 117491 388837 67 126232 420033 68 135500 453259 69 145379 488753 70 155790 526515 71 166875 566783 72 178567 609557 73 190982 655121 74 204127 703475 75 217928 754903 76 232651 809405 77 248060 867311 78 264409 928621 79 281641 993665 80 299733 1062443 81 318981 1135345 82 339049 1212371 83 360366 1293911 84 382622 1379965 85 406083 1470983 86 430873 1566965 87 456663 1668361 88 484013 1775171 89 512632 1887919 90 542499 2006605 91 574070 2131753 92 607011 2263363 93 641602 2402033 94 677880 2547763 95 715912 2701151 96 755684 2862197 97 797262 3031593 98 840860 3209339 99 886447 3396127 100 934154 3591957 (eof)