A334731 - OEIS

%I #20 Sep 08 2022 08:46:25

%S 1,1,1,1,1,12,1,1,1,2,1,48,1,4,3,1,1,36,1,4,1,4,1,576,1,2,1,224,1,

%T 5184,1,1,3,2,1,144,1,4,1,40,1,2304,1,16,9,4,1,2304,1,2,9,4,1,864,1,

%U 1792,1,2,1,995328,1,4,1,1,1,20736,1,4,3,128,1,5184,1,2

%N a(n) = Product_{d|n} gcd(sigma(d), pod(d)) where sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

%H Antti Karttunen, <a href="/A334731/b334731.txt">Table of n, a(n) for n = 1..16384</a>

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

%e a(6) = gcd(sigma(1), pod(1)) * gcd(sigma(2), pod(2)) * gcd(sigma(3), pod(3)) * gcd(sigma(6), pod(6)) = gcd(1, 1) * gcd(3, 2) * gcd(4, 3) * gcd(12, 36) = 1 * 1 * 1 * 12 = 12.

%t a[n_] := Product[GCD[DivisorSigma[1, 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) pod(n) = vecprod(divisors(n));

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

%Y Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(d), pod(d))).

%Y Cf. A000005 (tau(n)), A000203 (sigma(n)), A306682 (gcd(sigma(n), pod(n))).

%K nonn

%O 1,6

%A _Jaroslav Krizek_, May 09 2020