A324987 a(n) = Product_{d|n} (tau(d)*sigma(d)) where tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

1, 6, 8, 126, 12, 2304, 16, 7560, 312, 5184, 24, 8128512, 28, 9216, 9216, 1171800, 36, 21026304, 40, 27433728, 16384, 20736, 48, 234101145600, 1116, 28224, 49920, 65028096, 60, 110075314176, 64, 442940400, 36864, 46656, 36864, 60754075619328, 76, 57600, 50176

Offset: 1

Author

Jaroslav Krizek, Mar 23 2019

Keywords

• nonn

Comments

n divides a(n) for n: 1, 2, 6, 8, 12, 18, 24, 28, 36, 40, 48, 54, 56, 72, 80, 84, 96, 108, 112, 117, ...

Examples

			a(6) = (tau(1)*sigma(1)) * (tau(2)*sigma(2)) * (tau(3)*sigma(3)) * (tau(6)*sigma(6)) = (1*1) * (2*3) * (2*4) * (4*12) = 2304.
		

Crossrefs

Cf. A000005, A000203, A064840, A206032, A211776, A324986.

Programs

Magma

[&*[NumberOfDivisors(d) * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]]

Mathematica

Table[Product[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Mar 23 2019 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, numdiv(d[k])*sigma(d[k])); \\ Michel Marcus, Mar 23 2019

Formula

a(n) = Product_{d|n} tau(d) * Product_{d|n} sigma(d) = A211776(n) * A206032(n).

a(p) = 2*(p + 1) for p = primes (A000040).

a(n) = Product_{d|n} A064840(d). - Antti Karttunen, Mar 28 2019