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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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, 132, 156, 128, 17, 18, 44, 78, 168, 204, 312, 256, 19, 21, 52, 88, 198, 228, 408, 684, 512, 23, 22, 56, 102, 210, 264, 456, 696, 1020, 1024, 25, 24, 68, 104, 220, 276, 468, 744, 1140, 1380
Offset: 1

Views

Author

Omar E. Pol, May 22 2025

Keywords

Comments

This is a permutation of the positive integers.

Examples

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  20,  42,  84, 156, 312,  684, 1020, 1380, 1860, 3480, 3720,  4920, ...
      2,  5, 10,  28,  60, 132, 204, 408,  696, 1140, 1740, 2220, 3660, 4440,  5160, ...
      4,  7, 12,  30,  66, 168, 228, 456,  744, 1332, 2040, 2460, 4020, 5580,  5640, ...
      8,  9, 14,  40,  78, 198, 264, 468,  780, 1368, 2088, 2580, 4140, 6960,  6360, ...
     16, 11, 15,  44,  88, 210, 276, 510,  816, 1392, 2232, 2664, 4260, 7224,  6660, ...
     32, 13, 18,  52, 102, 220, 330, 552,  828, 1476, 2280, 2760, 4380, 7632,  7080, ...
     64, 17, 21,  56, 104, 234, 342, 570,  888, 1488, 2436, 2820, 4740, 7896,  7380, ...
    128, 19, 22,  68, 110, 252, 348, 612,  912, 1548, 2544, 2952, 4872, 8280,  7440, ...
    256, 23, 24,  70, 114, 260, 372, 624,  930, 1560, 2604, 3096, 4980, 8496,  7740, ...
    512, 25, 26,  76, 120, 272, 390, 660,  936, 1656, 2736, 3180, 5208, 8784,  8880, ...
   1024, 27, 33,  80, 126, 304, 396, 690,  984, 1692, 2790, 3384, 5220, 8904,  9912, ...
   2048, 29, 34,  90, 130, 306, 414, 792, 1032, 1710, 2832, 3420, 5256, 9030, 10248, ...
   4096, 31, 35,  92, 136, 336, 420, 870, 1044, 1776, 2928, 3540, 5328, 9324, 10440, ...
   8192, 37, 36,  99, 138, 340, 440, 920, 1104, 1908, 3060, 3612, 5340, 9648, 10512, ...
  16384, 41, 38, 100, 140, 368, 444, 966, 1110, 1932, 3108, 3816, 5520, 9660, 10836, ...
  ...
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.

Links

Crossrefs

Companion of A383961.
Row 1 gives A384232.
Column 1 gives A000079.
Cf. A006005, A027750, A061345, A065091, A087134.

Programs

  • Mathematica
    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 *)

Formula

Conjecture: T(n,2) = A061345(n).

A384231 Index of the largest odd noncomposite divisor in the list of divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 3, 1, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 4, 2, 4, 2, 1, 3, 3, 3, 3, 2, 3, 3, 4, 2, 5, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 3, 2, 5, 2, 3, 3, 1, 3, 5, 2, 4, 3, 4, 2, 3, 2, 3, 3, 4, 3, 5, 2, 4, 2, 3, 2, 6, 3, 3, 3, 5, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4
Offset: 1

Views

Author

Omar E. Pol, May 29 2025

Keywords

Comments

a(n) = 1 if and only if n is a power of 2.

Examples

			For n = 30 the divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30] and the largest odd noncomposite divisor is 5 and 5 is its 4th divisor, so a(30) = 4.

Crossrefs

Companion of A383401.
Right border of A384234.
Cf. A006005 (odd noncomposite numbers).
Cf. A000079, A027750, A384232, A384233.

Programs

  • Mathematica
    a[n_] := Module[{m = n/2^IntegerExponent[n, 2]}, If[m == 1, 1, Position[Divisors[n], FactorInteger[m][[-1, 1]]][[1, 1]]]]; Array[a, 100] (* Amiram Eldar, May 29 2025 *)

A384234 Irregular triangle read by rows: T(n,k) is the index of the k-th odd noncomposite divisor in the list of divisors of n, with n >=1, k >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 4, 1, 2, 1, 1, 2, 3, 1, 3, 1, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 1, 3, 5, 1, 2, 1, 4, 1, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3
Offset: 1

Views

Author

Omar E. Pol, May 29 2025

Keywords

Comments

Row n lists the indices of the odd noncomposite divisors in the list of divisors of n.
Row n is [1] if and only if n is a power of 2 (A000079).

Examples

			Triangle begins (n = 1..21):
  1;
  1;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2, 3;
  1;
  1, 2;
  1, 3;
  1, 2;
  1, 4;
  1, 2, 3;
  ...
For n = 30 the divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30] and the odd noncomposite divisors are [1, 3, 5] and the indices of them in the list of divisors are [1, 3, 4] respectively, so the 30th row of the triangle is [1, 3, 4].

Crossrefs

Companion of A383962.
Column 1 gives A000012.
Right border gives A384231.
Cf. A006005 (odd noncomposite numbers).
Cf. A000079, A027750, A384232, A384233.

Programs

  • Mathematica
    row[n_] := Module[{m = n/2^IntegerExponent[n, 2]}, Join[{1}, If[m == 1, {}, Position[Divisors[n], #] & /@ FactorInteger[m][[;; , 1]] // Flatten]]]; Array[row, 50] // Flatten (* Amiram Eldar, May 29 2025 *)
Showing 1-3 of 3 results.

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