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A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.

%I A384233 #17 May 29 2025 00:12:45
%S A384233 1,2,3,4,5,6,8,7,10,20,16,9,12,28,42,32,11,14,30,60,84,64,13,15,40,66,
%T A384233 132,156,128,17,18,44,78,168,204,312,256,19,21,52,88,198,228,408,684,
%U A384233 512,23,22,56,102,210,264,456,696,1020,1024,25,24,68,104,220,276,468,744,1140,1380
%N A384233 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd noncomposite divisor is its k-th divisor, n >= 1, k >= 1.
%C A384233 This is a permutation of the positive integers.
%H A384233 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A384233 Conjecture: T(n,2) = A061345(n).
%e A384233 The corner 15 X 15 of the square array is as follows:
%e A384233       1,  3,  6,  20,  42,  84, 156, 312,  684, 1020, 1380, 1860, 3480, 3720,  4920, ...
%e A384233       2,  5, 10,  28,  60, 132, 204, 408,  696, 1140, 1740, 2220, 3660, 4440,  5160, ...
%e A384233       4,  7, 12,  30,  66, 168, 228, 456,  744, 1332, 2040, 2460, 4020, 5580,  5640, ...
%e A384233       8,  9, 14,  40,  78, 198, 264, 468,  780, 1368, 2088, 2580, 4140, 6960,  6360, ...
%e A384233      16, 11, 15,  44,  88, 210, 276, 510,  816, 1392, 2232, 2664, 4260, 7224,  6660, ...
%e A384233      32, 13, 18,  52, 102, 220, 330, 552,  828, 1476, 2280, 2760, 4380, 7632,  7080, ...
%e A384233      64, 17, 21,  56, 104, 234, 342, 570,  888, 1488, 2436, 2820, 4740, 7896,  7380, ...
%e A384233     128, 19, 22,  68, 110, 252, 348, 612,  912, 1548, 2544, 2952, 4872, 8280,  7440, ...
%e A384233     256, 23, 24,  70, 114, 260, 372, 624,  930, 1560, 2604, 3096, 4980, 8496,  7740, ...
%e A384233     512, 25, 26,  76, 120, 272, 390, 660,  936, 1656, 2736, 3180, 5208, 8784,  8880, ...
%e A384233    1024, 27, 33,  80, 126, 304, 396, 690,  984, 1692, 2790, 3384, 5220, 8904,  9912, ...
%e A384233    2048, 29, 34,  90, 130, 306, 414, 792, 1032, 1710, 2832, 3420, 5256, 9030, 10248, ...
%e A384233    4096, 31, 35,  92, 136, 336, 420, 870, 1044, 1776, 2928, 3540, 5328, 9324, 10440, ...
%e A384233    8192, 37, 36,  99, 138, 340, 440, 920, 1104, 1908, 3060, 3612, 5340, 9648, 10512, ...
%e A384233   16384, 41, 38, 100, 140, 368, 444, 966, 1110, 1932, 3108, 3816, 5520, 9660, 10836, ...
%e A384233   ...
%e A384233 The divisors of 42 are [1, 2, 3, 6, 7, 14, 21, 42] and the largest odd noncomposite divisor is 7 and 7 is its 5th divisor, so T(1,5) = 42 because 42 the smallest number having that property.
%t A384233 f[n_] := FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]]; seq[m_] := Module[{t = Table[0, {m}, {m}], v = Table[0, {m}], c = 0, k = 1, i, j}, While[c < m*(m + 1)/2, i = f[k]; If[i <= m, j = v[[i]] + 1; If[j <= m - i + 1, t[[i]][[j]] = k; v[[i]]++; c++]]; k++]; Table[t[[j]][[i - j + 1]], {i, 1, m}, {j, 1, i}] // Flatten]; seq[11] (* _Amiram Eldar_, May 23 2025 *)
%Y A384233 Companion of A383961.
%Y A384233 Row 1 gives A384232.
%Y A384233 Column 1 gives A000079.
%Y A384233 Cf. A006005, A027750, A061345, A065091, A087134.
%K A384233 nonn,tabl
%O A384233 1,2
%A A384233 _Omar E. Pol_, May 22 2025