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.
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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 4, 3, 4, 1, 6, 1, 1, 1, 1, 1, 12
Offset: 0
Examples
Triangle starts; [0] 0; [1] 1, 1; [2] 1, 1, 2; [3] 1, 1, 1, 3; [4] 1, 1, 4, 1, 4; [5] 1, 1, 1, 1, 1, 5; [6] 1, 1, 2, 3, 1, 1, 6; [7] 1, 1, 1, 1, 1, 1, 1, 7; [8] 1, 1, 8, 1, 4, 1, 1, 1, 8; [9] 1, 1, 1, 9, 1, 1, 1, 1, 1, 9; [10] 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 10;
Links
- Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (Nov. 2000), 783-799.
- Jeffrey C. Lagarias and Wijit Yangjit, The factorial function and generalizations, extended, arXiv:2310.12949 [math.NT], 2023. See Section 7.2 pp. 20-21 and Table 1 p. 29.
Programs
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Maple
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);
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Mathematica
T[n_, 0] := If[n == 0, 0, 1]; T[n_, 1] := 1; T[n_, k_] := k^IntegerExponent[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // MatrixForm