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A334729 a(n) = Product_{d|n} gcd(tau(d), sigma(d)).

%I A334729 #18 Sep 08 2022 08:46:25
%S A334729 1,1,2,1,2,8,2,1,2,4,2,16,2,8,16,1,2,24,2,24,16,8,2,64,2,4,8,16,2,
%T A334729 1024,2,3,16,4,16,48,2,8,16,48,2,2048,2,48,96,8,2,128,6,12,16,8,2,768,
%U A334729 16,128,16,4,2,147456,2,8,32,3,16,2048,2,24,16,1024,2
%N A334729 a(n) = Product_{d|n} gcd(tau(d), sigma(d)).
%H A334729 Robert Israel, <a href="/A334729/b334729.txt">Table of n, a(n) for n = 1..10000</a>
%F A334729 a(p) = 2 for p = odd primes (A065091).
%e A334729 a(6) = gcd(tau(1), sigma(1)) * gcd(tau(2), sigma(2)) * gcd(tau(3), sigma(3)) * gcd(tau(6), sigma(6)) = gcd(1, 1) * gcd(2, 3) * gcd(2, 4) * gcd(4, 12) = 1 * 1 * 2 * 4 = 8.
%p A334729 g:= proc(d) option remember; igcd(numtheory:-tau(d), numtheory:-sigma(d)) end proc:
%p A334729 f:= n -> mul(g(d), d = numtheory:-divisors(n)):
%p A334729 map(f, [$1..100]); # _Robert Israel_, May 11 2020
%t A334729 a[n_] := Product[GCD[DivisorSigma[0, d], DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 100] (* _Amiram Eldar_, May 09 2020 *)
%o A334729 (Magma) [&*[GCD(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
%o A334729 (PARI) a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), sigma(d[k]))); \\ _Michel Marcus_, May 09-11 2020
%Y A334729 Cf. A334491 (Product_{d|n} gcd(d, sigma(d))), A334579 (Sum_{d|n} gcd(tau(d), sigma(d))).
%Y A334729 Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).
%K A334729 nonn
%O A334729 1,3
%A A334729 _Jaroslav Krizek_, May 09 2020