A382944 Table read by rows: T(n, k) = valuation(n, k) for k >= 2, 1 for k = 1 and 0^n for k = 0.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Peter Luschny, Apr 09 2025

Keywords

• nonn
• tabl

Comments

If a term T(n, k) > 1 is replaced by 1 the triangle reduces to the divisibility triangle A113704. In addition to divisibility, T(n, k) indicates the order of divisibility. For n, k >= 2 this is defined as the multiplicity of a divisor, i.e., the exponent of the highest order of k that divides n. For a prime number p T(n, p) is called p-adic valuation or p-adic order of n. See also the comments in A382883.

Examples

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1, 1;
  [3] 0, 1, 0, 1;
  [4] 0, 1, 2, 0, 1;
  [5] 0, 1, 0, 0, 0, 1;
  [6] 0, 1, 1, 1, 0, 0, 1;
  [7] 0, 1, 0, 0, 0, 0, 0, 1;
  [8] 0, 1, 3, 0, 1, 0, 0, 0, 1;
  [9] 0, 1, 0, 2, 0, 0, 0, 0, 0, 1;
		

Crossrefs

Cf. A169594 (row sums), A113704, A382881 (inverse), A382883, A057427.

Programs

Maple

A382944 := proc(n, k) if k = 0 then 0^n elif k = 1 then 1 else padic:-ordp(n, k) fi end: seq(seq(A382944(n, k), k = 0..n), n = 0..12);

Mathematica

T[n_, 0] := T[n, 0] = Boole[n == 0]; T[n_, 1] := T[n, 1] = 1; T[n_, k_] := T[n, k] = IntegerExponent[n, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2025 *)

SageMath

@cached_function
def A382944(n: int, k: int) -> int:
    if not ZZ(k).divides(n) or k > n: return 0
    if k == n or k == 1: return 1
    return valuation(n, k)
for n in range(13): print([n], [A382944(n, k) for k in range(n + 1)])

Formula

A113704(n, k) = A057427(T(n, k)). - Amiram Eldar, Apr 29 2025