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a(n) = Product_{d|n} gcd(d, sigma(d)).
5

%I #24 Sep 08 2022 08:46:25

%S 1,1,1,1,1,6,1,1,1,2,1,24,1,2,3,1,1,18,1,4,1,2,1,288,1,2,1,56,1,216,1,

%T 1,3,2,1,72,1,2,1,40,1,72,1,8,9,2,1,1152,1,2,3,4,1,108,1,448,1,2,1,

%U 20736,1,2,1,1,1,216,1,4,3,8,1,2592,1,2,3,8,1,72

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

%H Antti Karttunen, <a href="/A334491/b334491.txt">Table of n, a(n) for n = 1..16384</a>

%F a(p) = 1 for p = primes (A000040).

%F a(n) = Product_{d|n} A009194(d). - _Antti Karttunen_, May 09 2020

%e a(6) = gcd(1, sigma(1)) * gcd(2, sigma(2)) * gcd(3, sigma(3)) * gcd(6, sigma(6)) = gcd(1, 1) * gcd(2, 3) * gcd(3, 4) * gcd(6, 12) = 1 * 1 * 1 * 6 = 6.

%t a[n_] := Product[GCD[d, DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 80] (* _Amiram Eldar_, May 03 2020 *)

%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], sigma(d[k]))); \\ _Michel Marcus_, May 03-11 2020

%Y Cf. A334490 (Sum_{d|n} gcd(d, sigma(d))), A007955 (pod(n) = Product_{d|n} gcd(d, pod(d))).

%Y Cf. A000040, A000203 (sigma(n)), A009194.

%K nonn

%O 1,6

%A _Jaroslav Krizek_, May 03 2020