A072046 Greatest common divisor of product of divisors of n and product of non-divisors < n.

1, 1, 1, 1, 1, 4, 1, 2, 3, 4, 1, 144, 1, 4, 45, 32, 1, 72, 1, 320, 63, 4, 1, 82944, 125, 4, 729, 448, 1, 162000, 1, 32768, 99, 4, 1225, 3359232, 1, 4, 117, 2560000, 1, 63504, 1, 704, 91125, 4, 1, 254803968, 343, 125000, 153, 832, 1, 8503056, 3025, 9834496, 171, 4, 1

Offset: 1

Author

Reinhard Zumkeller, Jul 29 2002

Keywords

• nonn
• nice

Examples

			a(12) = GCD(A007955(12), A055067(12)) = GCD(1*2*3*4*6*12,5*7*8*9*10*11) = GCD(1728,277200) = 144;
a(13) = GCD(A007955(13), A055067(13)) = GCD(1*13,2*3*4*5*6*7*8*9*10*11*12) = GCD(13,479001600) = 1.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007955, A055067.

Programs

Haskell

a072046 n = gcd (a007955 n) (a055067 n)
-- Reinhard Zumkeller, Feb 06 2012

Mathematica

a[n_] := (dd = Divisors[n]; GCD[Times @@ dd, Times @@ Complement[Range[n], dd]]); Array[a, 59]
a[n_] := GCD[(p = n^(DivisorSigma[0, n]/2)), n!/p]; Array[a, 60] (* Amiram Eldar, Jun 26 2022 *)

Python

from math import isqrt, gcd, factorial
from sympy import divisor_count
def A072046(n): return gcd(p:=isqrt(n)**c if (c:=divisor_count(n)) & 1 else n**(c//2),factorial(n)//p) # Chai Wah Wu, Jun 25 2022

Formula

a(n) = GCD(A007955(n), A055067(n)).