A381885 - OEIS

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.

%I #13 Apr 01 2025 08:55:35

%S 1,1,1,4,1,6,1,32,9,10,1,288,1,14,15,2048,1,972,1,800,21,22,1,55296,

%T 25,26,243,1568,1,27000,1,65536,33,34,35,10077696,1,38,39,256000,1,

%U 74088,1,3872,6075,46,1,169869312,49,12500,51,5408,1,1417176,55,702464,57

%N 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_valuation">p-adic valuation</a>.

%F If the base of the factors of the product is restricted to prime numbers then A005451 is generated.

%F a(p) = 1 if p is prime.

%F a(n) = A364813(n) / n.

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

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

%o (PARI) a(n) = prod(k=2, n-1, k^valuation(n, k)); \\ _Michel Marcus_, Apr 01 2025

%Y Cf. A364813, A005451.

%K nonn

%O 1,4

%A _Peter Luschny_, Apr 01 2025