A364813 -id:A364813 - cp's OEIS frontend

A381885 a(n) = Product_{k=2..n-1} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise is the exponent of the highest power of k that divides n.

1, 1, 1, 4, 1, 6, 1, 32, 9, 10, 1, 288, 1, 14, 15, 2048, 1, 972, 1, 800, 21, 22, 1, 55296, 25, 26, 243, 1568, 1, 27000, 1, 65536, 33, 34, 35, 10077696, 1, 38, 39, 256000, 1, 74088, 1, 3872, 6075, 46, 1, 169869312, 49, 12500, 51, 5408, 1, 1417176, 55, 702464, 57

Offset: 1

Peter Luschny, Apr 01 2025

nonn

Wikipedia, p-adic valuation.

Cf. A364813, A005451.

with(padic): a := n -> local k; mul(k^ordp(n, k), k = 2.. n-1): seq(a(n), n = 1..57);

Table[Product[k^IntegerExponent[n, k], {k, 2, n - 1}], {n, 1, 57}]

a(n) = prod(k=2, n-1, k^valuation(n, k)); \\ Michel Marcus, Apr 01 2025

If the base of the factors of the product is restricted to prime numbers then A005451 is generated.
a(p) = 1 if p is prime.
a(n) = A364813(n) / 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.

Original entry on oeis.org

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

Peter Luschny, Apr 07 2025

			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;

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.

Cf. A364813 (row product), A381886, A286563.

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[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

cp's OEIS Frontend

A381885 a(n) = Product_{k=2..n-1} k^ord(n, k) where ord(n, k) = 0 if k does not divide n, otherwise is the exponent of the highest power of k that divides n.

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