A307101 a(n) = Product_{d|n} (tau(d)*pod(d)) where tau(k) = the number of divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

1, 4, 6, 96, 10, 3456, 14, 24576, 486, 16000, 22, 859963392, 26, 43904, 54000, 125829120, 34, 9795520512, 38, 18432000000, 148176, 170368, 46, 584325558976905216, 3750, 281216, 1417176, 138784407552, 58, 80621568000000000, 62, 24739011624960, 574992, 628864

Offset: 1

Author

Jaroslav Krizek, Apr 25 2019

Keywords

• nonn

Comments

n divides a(n) for all n.

Examples

			a(6) = (tau(1)*pod(1)) * (tau(2)*pod(2)) * (tau(3)*pod(3)) * (tau(6)*pod(6)) = (1*1) * (2*2) * (2*3) * (4*36) = 3456.
		

Crossrefs

Cf. A000005, A007955, A307100.

Programs

Magma

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

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, my(dd=divisors(d[k])); #dd*vecprod(dd)); \\ Michel Marcus, Apr 25 2019

Formula

a(n) = Product_{d|n} tau(d) * Product_{d|n} pod(d) = A211776(n) * A266265(n).

a(p) = 2p for p = primes (A000040).