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.

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

Author

Omar E. Pol, May 26 2025

Keywords

• nonn
• tabf

Comments

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).

Examples

			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].
		

Crossrefs

Column 1 gives A000012.

Row lengths gives A001227.

Right border gives A383401.

Cf. A000005, A000027, A000079, A027750, A065091, A174090, A182469.

Programs

Mathematica

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