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.

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

1,4

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).

Links

Table of n, a(n) for n=1..102.

Example

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].

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 *)

Crossrefs

Companion of A383962.

Column 1 gives A000012.

Right border gives A384231.

Cf. A006005 (odd noncomposite numbers).

Cf. A000079, A027750, A384232, A384233.

Sequence in context: A296559 A233548 A080027 * A383962 A341970 A375001

Adjacent sequences: A384231 A384232 A384233 * A384235 A384236 A384237

Keyword

nonn,tabf,easy

Author

Omar E. Pol, May 29 2025

Status

approved