A383401 -id:A383401 - cp's OEIS frontend

A383402 Smallest number whose largest odd divisor is its n-th divisor.

1, 3, 6, 15, 18, 36, 30, 105, 60, 120, 90, 315, 816, 1360, 180, 700, 450, 360, 720, 1008, 420, 1540, 630, 900, 840, 1080, 1620, 1680, 2160, 1800, 1890, 5280, 1260, 3240, 3150, 17325, 7200, 29120, 5670, 9072, 2520, 3960, 10296, 18144, 3780, 20020, 5040, 7920, 10800

Offset: 1

Omar E. Pol, May 14 2025

nonn

From Peter Munn, May 15 2025 and May 20 2025: (Start)
A038547 is easily seen to be an upper bound for the sequence and a term equals this upper bound if and only if it is odd. Moreover, if a(n) = 2m with m odd, then the largest odd divisor of 2m is m, its second largest divisor, and a(n) = 2 * A038547((n+1)/2). It follows that 1 is the only term not divisible by 4 or by a nonunit term of A038547.
a(8) = 105 is the last squarefree term. (This is a corollary to lemma: prime p > 9 cannot be a divisor of a squarefree term. Proof of lemma: Let p divide squarefree k. If 3p is also divisor, set m = 9k/p, otherwise set m = 3k/p. Then k is not a term as m is a smaller number whose largest odd divisor is in the same position in the divisor list.)
(End)
If a(n) = m then m has at least n divisors. - David A. Corneth, May 16 2025
Every term a(n) = t > 1 is divisible by 2 or 3. Proof: Suppose it is not. Then it is odd and n is the number of divisors of t (cf. A000005). But t is not the smallest number that has n odd divisors that is odd. Setting every prime factor p to the largest prime < p and then multiplying gives a smaller odd number that has n divisors (cf. A064989). - David A. Corneth, May 17 2025

			The divisors of 18 are [1, 2, 3, 6, 9, 18] and the largest odd divisor is 9 and 9 is its 5th divisor, so a(5) = 18 because 18 the smallest number having that property.

David A. Corneth, Table of n, a(n) for n = 1..872
David A. Corneth, PARI program
David A. Corneth, Upper bounds on a(n) for n = 1..10000
Michael De Vlieger, Prime Power Decomposition of A383402(n), n = 1..261.

Row 1 of A383961.
The range of terms is a subset of {1} U A355200.
Cf. A000005, A000265, A001227, A005117, A027750, A038547, A064989, A182469, A383401.
See A221647 for other sequences giving the smallest number whose n-th divisor satisfies some condition.

With[{t = Table[If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[1]]], {n, 1, 30000}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 14 2025 *)

a(n) = my(k=1); while (select(x->(x==k/2^valuation(k,2)), divisors(k), 1)[1] != n, k++); k; \\ Michel Marcus, May 14 2025

PARI
```
\\ See Corneth link
```

a(n) = min({k : A000005(k) >= n & A027750(k,n) = A000265(k)}). - Peter Munn, May 14 2025

More terms from Amiram Eldar, May 14 2025

A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 15, 16, 11, 10, 20, 18, 32, 13, 12, 21, 50, 36, 64, 17, 14, 27, 81, 45, 30, 128, 19, 22, 28, 88, 63, 42, 105, 256, 23, 24, 33, 98, 75, 54, 135, 60, 512, 29, 25, 35, 104, 99, 66, 165, 84, 120, 1024, 31, 26, 39, 136, 117, 70, 189, 108, 140, 90

Offset: 1

Omar E. Pol, May 16 2025

This is a permutation of the positive integers.
From Peter Munn, May 18 2025: (Start)
Numbers with the same factorization pattern of their sequence of divisors (see A290110) and the same parity appear here in the same column.
For example, each column k > 2 includes the subsequence 2^(k-2) * p for all prime p > 2^(k-2).
(End)

			The corner 15 X 15 of the square array is as follows:
      1,  3,  6,  15,  18,  36,  30, 105,  60, 120,  90, 315,  816, 1360, 180, ...
      2,  5,  9,  20,  50,  45,  42, 135,  84, 140, 126, 324,  880, 1520, 210, ...
      4,  7, 10,  21,  81,  63,  54, 165, 108, 168, 150, 432,  912, 1632, 252, ...
      8, 11, 12,  27,  88,  75,  66, 189, 132, 220, 198, 440, 1040, 1760, 270, ...
     16, 13, 14,  28,  98,  99,  70, 195, 156, 240, 216, 495, 1056, 1824, 300, ...
     32, 17, 22,  33, 104, 117,  72, 200, 162, 260, 234, 520, 1104, 1840, 330, ...
     64, 19, 24,  35, 136, 147,  78, 231, 204, 308, 264, 525, 1120, 1904, 378, ...
    128, 23, 25,  39, 152, 153, 100, 255, 225, 340, 280, 528, 1144, 2000, 390, ...
    256, 29, 26,  40, 176, 171, 102, 273, 228, 364, 294, 560, 1232, 2080, 396, ...
    512, 31, 34,  44, 184, 175, 110, 285, 276, 380, 306, 585, 1248, 2128, 462, ...
   1024, 37, 38,  51, 208, 207, 114, 297, 348, 405, 312, 616, 1392, 2208, 468, ...
   2048, 41, 46,  52, 232, 243, 130, 345, 372, 460, 336, 624, 1456, 2288, 510, ...
   4096, 43, 48,  55, 242, 245, 138, 351, 400, 476, 342, 675, 1458, 2320, 546, ...
   8192, 47, 49,  56, 248, 261, 144, 357, 441, 480, 350, 680, 1488, 2464, 570, ...
  16384, 53, 58,  57, 296, 272, 154, 375, 444, 500, 408, 693, 1496, 2480, 588, ...
  ...

Column 1 gives A000079.
Column 2 gives A065091.
Column 3 consists of (A001248 U A091629 U A100484)\{4}.
Column 4 consists of numbers >= 15 in (A001749 U A030078 U A046388 U A070875).
Row 1 gives A383402.
Cf. A000005, A000265, A001227, A027750, A038547, A174090, A182469, A290110, A383401.

f[n_] := If[OddQ[n], DivisorSigma[0, n], FirstPosition[Divisors[n], n/2^IntegerExponent[n, 2]][[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 16 2025 *)

A383962 Irregular triangle read by rows: T(n,k) is the index of the k-th odd 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, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 4, 1, 1, 2, 1, 3, 5, 1, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 2, 1, 3, 1, 2, 3, 1, 3, 1, 2, 3, 4, 1, 4, 1, 2, 1, 3, 4, 7, 1, 2, 1, 1, 2, 3, 4, 1, 3, 1, 2, 3, 4, 1, 3, 6, 1, 2, 1, 3, 1, 2, 3, 4, 1, 4, 1, 2, 1, 3, 5, 7, 1, 2, 1, 4, 1, 2, 3, 4, 5, 6

Offset: 1

Omar E. Pol, May 26 2025

Row n lists the indices of the odd divisors in the list of divisors of n.
If n is odd then row n lists the first A000005(n) positive integers (A000027).
Row n is [1] if and only if n is a power of 2 (A000079).
Row n is [1, 2] if and only if n is an odd prime (A065091).

			Triangle begins (n = 1..21):
  1;
  1;
  1, 2;
  1;
  1, 2;
  1, 3;
  1, 2;
  1;
  1, 2, 3;
  1, 3;
  1, 2;
  1, 3;
  1, 2;
  1, 3;
  1, 2, 3, 4;
  1;
  1, 2;
  1, 3, 5;
  1, 2;
  1, 4;
  1, 2, 3, 4;
  ...
For n = 20 the divisors of 20 are [1, 2, 4, 5, 10, 20]. The odd divisors are [1, 5] and their indices in the list of divisors are [1, 4] respectively, so the 20th row of the triangle is [1, 4].

Column 1 gives A000012.
Row lengths gives A001227.
Right border gives A383401.
Cf. A000005, A000027, A000079, A027750, A065091, A174090, A182469.

row[n_] := Position[Divisors[n], ?OddQ] // Flatten; Array[row, 45] // Flatten (* _Amiram Eldar, May 26 2025 *)

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

Omar E. Pol, May 29 2025

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

			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.

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

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 *)

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A383402 Smallest number whose largest odd divisor is its n-th divisor.

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A383961 Square array read by upward antidiagonals: T(n,k) is the n-th number whose largest odd divisor is its k-th divisor, n >= 1, k >= 1.

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A383962 Irregular triangle read by rows: T(n,k) is the index of the k-th odd divisor in the list of divisors of n, with n >= 1, k >= 1.

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

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