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a(n) = Product_{d|n} gcd(d, tau(d)).
2

%I #18 Mar 30 2024 15:07:08

%S 1,2,1,2,1,4,1,8,3,4,1,24,1,4,1,8,1,72,1,8,1,4,1,768,1,4,3,8,1,16,1,

%T 16,1,4,1,3888,1,4,1,256,1,16,1,8,9,4,1,1536,1,8,1,8,1,144,1,256,1,4,

%U 1,2304,1,4,9,16,1,16,1,8,1,16,1,1492992,1,4,3,8,1

%N a(n) = Product_{d|n} gcd(d, tau(d)).

%H Antti Karttunen, <a href="/A334664/b334664.txt">Table of n, a(n) for n = 1..10000</a>

%F a(p) = 1 for p = odd primes (A065091).

%e a(6) = gcd(1, tau(1)) * gcd(2, tau(2)) * gcd(3, tau(3)) * gcd(6, tau(6)) = gcd(1, 1) * gcd(2, 2) * gcd(3, 2) * gcd(6, 4) = 1 * 2 * 1 * 2 = 4.

%t Table[Times@@GCD[Divisors[n],DivisorSigma[0,Divisors[n]]],{n,80}] (* _Harvey P. Dale_, Mar 30 2024 *)

%o (Magma) [&*[GCD(d, #Divisors(d)): d in Divisors(n)]: n in [1..100]]

%o (PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(d[k], numdiv(d[k]))); \\ _Michel Marcus_, May 08-11 2020

%Y Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).

%Y Cf. A000005 (tau(n)), A000203 (sigma(n)), A009191 (gcd(n, tau(n))).

%K nonn

%O 1,2

%A _Jaroslav Krizek_, May 07 2020