A334730 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).

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, 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, 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

Offset: 1

Author

Jaroslav Krizek, May 09 2020

Keywords

• nonn

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

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

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

Programs

Magma

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

Mathematica

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

PARI

pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

Formula

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