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

%I A382944 #17 Apr 30 2025 03:35:51
%S A382944 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,
%T A382944 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,
%U A382944 0,0,0,0,0,0,0,0,0,1,0,1,2,1,1,0,1,0,0,0,0,0,1
%N A382944 Table read by rows: T(n, k) = valuation(n, k) for k >= 2, 1 for k = 1 and 0^n for k = 0.
%C A382944 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.
%F A382944 A113704(n, k) = A057427(T(n, k)). - _Amiram Eldar_, Apr 29 2025
%e A382944 Triangle starts:
%e A382944   [0] 1;
%e A382944   [1] 0, 1;
%e A382944   [2] 0, 1, 1;
%e A382944   [3] 0, 1, 0, 1;
%e A382944   [4] 0, 1, 2, 0, 1;
%e A382944   [5] 0, 1, 0, 0, 0, 1;
%e A382944   [6] 0, 1, 1, 1, 0, 0, 1;
%e A382944   [7] 0, 1, 0, 0, 0, 0, 0, 1;
%e A382944   [8] 0, 1, 3, 0, 1, 0, 0, 0, 1;
%e A382944   [9] 0, 1, 0, 2, 0, 0, 0, 0, 0, 1;
%p A382944 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);
%t A382944 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 *)
%o A382944 (SageMath)
%o A382944 @cached_function
%o A382944 def A382944(n: int, k: int) -> int:
%o A382944     if not ZZ(k).divides(n) or k > n: return 0
%o A382944     if k == n or k == 1: return 1
%o A382944     return valuation(n, k)
%o A382944 for n in range(13): print([n], [A382944(n, k) for k in range(n + 1)])
%Y A382944 Cf. A169594 (row sums), A113704, A382881 (inverse), A382883, A057427.
%K A382944 nonn,tabl
%O A382944 0,13
%A A382944 _Peter Luschny_, Apr 09 2025