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A382882 Triangle read by rows: T(n, k) = k^ord(n, k) where ord(n, k) is the p-adic order if n and k >= 2, 1 if k = 1, and 0^n if k = 0.

%I A382882 #7 Apr 08 2025 08:49:13
%S A382882 0,1,1,1,1,2,1,1,1,3,1,1,4,1,4,1,1,1,1,1,5,1,1,2,3,1,1,6,1,1,1,1,1,1,
%T A382882 1,7,1,1,8,1,4,1,1,1,8,1,1,1,9,1,1,1,1,1,9,1,1,2,1,1,5,1,1,1,1,10,1,1,
%U A382882 1,1,1,1,1,1,1,1,1,11,1,1,4,3,4,1,6,1,1,1,1,1,12
%N A382882 Triangle read by rows: T(n, k) = k^ord(n, k) where ord(n, k) is the p-adic order if n and k >= 2, 1 if k = 1, and 0^n if k = 0.
%H A382882 Manjul Bhargava, <a href="http://dx.doi.org/10.2307/2695734">The factorial function and generalizations</a>, Amer. Math. Monthly, 107 (Nov. 2000), 783-799.
%H A382882 Jeffrey C. Lagarias and Wijit Yangjit, <a href="https://arxiv.org/abs/2310.12949">The factorial function and generalizations, extended</a>, arXiv:2310.12949 [math.NT], 2023. See Section 7.2 pp. 20-21 and Table 1 p. 29.
%e A382882 Triangle starts;
%e A382882    [0] 0;
%e A382882    [1] 1, 1;
%e A382882    [2] 1, 1, 2;
%e A382882    [3] 1, 1, 1, 3;
%e A382882    [4] 1, 1, 4, 1, 4;
%e A382882    [5] 1, 1, 1, 1, 1, 5;
%e A382882    [6] 1, 1, 2, 3, 1, 1, 6;
%e A382882    [7] 1, 1, 1, 1, 1, 1, 1, 7;
%e A382882    [8] 1, 1, 8, 1, 4, 1, 1, 1, 8;
%e A382882    [9] 1, 1, 1, 9, 1, 1, 1, 1, 1, 9;
%e A382882   [10] 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 10;
%p A382882 ord := proc(n, d) if d = 1 then 1 elif d = 0 then ifelse(n = 0, 1, 0) else padic:-ordp(n, d) fi end: Trow := n -> local k; seq(k^ord(n, k), k = 0..n): seq(Trow(n), n = 0..12);
%t A382882 T[n_, 0] := If[n == 0, 0, 1]; T[n_, 1] := 1; T[n_, k_] := k^IntegerExponent[n, k];
%t A382882 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // MatrixForm
%Y A382882 Cf. A364813 (row product), A381886, A286563.
%K A382882 nonn,tabl
%O A382882 0,6
%A A382882 _Peter Luschny_, Apr 07 2025