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

%I A072046 #22 Jun 26 2022 02:11:28
%S A072046 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,
%T A072046 729,448,1,162000,1,32768,99,4,1225,3359232,1,4,117,2560000,1,63504,1,
%U A072046 704,91125,4,1,254803968,343,125000,153,832,1,8503056,3025,9834496,171,4,1
%N A072046 Greatest common divisor of product of divisors of n and product of non-divisors < n.
%H A072046 Reinhard Zumkeller, <a href="/A072046/b072046.txt">Table of n, a(n) for n = 1..10000</a>
%F A072046 a(n) = GCD(A007955(n), A055067(n)).
%e A072046 a(12) = GCD(A007955(12), A055067(12)) = GCD(1*2*3*4*6*12,5*7*8*9*10*11) = GCD(1728,277200) = 144;
%e A072046 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.
%t A072046 a[n_] := (dd = Divisors[n]; GCD[Times @@ dd, Times @@ Complement[Range[n], dd]]); Array[a, 59]
%t A072046 a[n_] := GCD[(p = n^(DivisorSigma[0, n]/2)), n!/p]; Array[a, 60] (* _Amiram Eldar_, Jun 26 2022 *)
%o A072046 (Haskell)
%o A072046 a072046 n = gcd (a007955 n) (a055067 n)
%o A072046 -- _Reinhard Zumkeller_, Feb 06 2012
%o A072046 (Python)
%o A072046 from math import isqrt, gcd, factorial
%o A072046 from sympy import divisor_count
%o A072046 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
%Y A072046 Cf. A007955, A055067.
%K A072046 nonn,nice
%O A072046 1,6
%A A072046 _Reinhard Zumkeller_, Jul 29 2002