A384233 -id:A384233 - cp's OEIS frontend

A384232 Smallest number whose largest odd noncomposite divisor is its n-th divisor.

1, 3, 6, 20, 42, 84, 156, 312, 684, 1020, 1380, 1860, 3480, 3720, 4920, 7320, 10980, 14640, 16920, 21960, 26280, 34920, 45720, 59640, 69840, 89880, 106680, 125160, 145320, 177240, 213360, 244440, 269640, 354480, 320040, 375480, 435960, 456120, 531720, 647640, 708120

Offset: 1

Omar E. Pol, May 23 2025

nonn

This coincide with A087134 except for the second term because here a(2) = 3 and there A087134(2) = 2.

			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 a(5) = 42 because 42 the smallest number having that property.

Row 1 of A384233.
Companion of A383402.
Cf. A006005, A027750, A087134.

With[{t = Table[FirstPosition[Divisors[n], FactorInteger[n/2^IntegerExponent[n, 2]][[-1, 1]]][[1]], {n, 1, 10^6}]}, TakeWhile[FirstPosition[t, #] & /@ Range[Max[t]] // Flatten, ! MissingQ[#] &]] (* Amiram Eldar, May 23 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 *)

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

Omar E. Pol, May 29 2025

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).

			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].

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

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 *)

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A384232 Smallest number whose largest odd noncomposite divisor is its n-th divisor.

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

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Mathematica

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.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica