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

Author

Peter Luschny, Apr 01 2025

Keywords

• nonn

Links

• Wikipedia, p-adic valuation.

Crossrefs

Cf. A364813, A005451.

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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.