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a(n) = Product_{d|n} gcd(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).
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%I #20 Sep 08 2022 08:46:25

%S 1,2,1,2,1,8,1,8,3,8,1,48,1,8,1,8,1,144,1,16,1,8,1,1536,1,8,3,16,1,

%T 256,1,16,1,8,1,7776,1,8,1,512,1,256,1,16,9,8,1,3072,1,16,1,16,1,1152,

%U 1,512,1,8,1,36864,1,8,9,16,1,256,1,16,1,256,1,2985984,1,8,3,16,1,256,1

%N a(n) = Product_{d|n} gcd(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

%H Antti Karttunen, <a href="/A334730/b334730.txt">Table of n, a(n) for n = 1..16384</a>

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

%e a(6) = gcd(tau(1), pod(1)) * gcd(tau(2), pod(2)) * gcd(tau(3), pod(3)) * gcd(tau(6), pod(6)) = gcd(1, 1) * gcd(2, 2) * gcd(2, 3) * gcd(4, 36) = 1 * 2 * 1 * 4 = 8.

%t a[n_] := Product[GCD[DivisorSigma[0, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* _Amiram Eldar_, May 09 2020 *)

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

%o (PARI)

%o pod(n) = vecprod(divisors(n));

%o a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), pod(d[k]))); \\ _Michel Marcus_, May 09-11 2020

%Y Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334662 (Sum_{d|n} gcd(tau(d), pod(d))).

%Y Cf. A000005 (tau(n)), A007955 (pod(n)), A306671 (gcd(tau(n), pod(n))).

%K nonn

%O 1,2

%A _Jaroslav Krizek_, May 09 2020