A334795 a(n) = Product_{d|n} lcm(d, tau(d)) where tau(k) is the number of divisors of k (A000005).

1, 2, 6, 24, 10, 144, 14, 192, 54, 400, 22, 20736, 26, 784, 3600, 15360, 34, 23328, 38, 288000, 7056, 1936, 46, 3981312, 750, 2704, 5832, 790272, 58, 207360000, 62, 1474560, 17424, 4624, 19600, 120932352, 74, 5776, 24336, 92160000, 82, 796594176, 86, 3066624

Offset: 1

Author

Jaroslav Krizek, May 12 2020

Keywords

• nonn
• look

Comments

From Robert Israel, Jun 25 2020: (Start)

If p is an odd prime, a(p) = 2*p.

If p is a prime > 3, a(p^2) = 6*p^3.

If p and q are distinct odd primes, a(p*q) = 16*p^2*q^2. (End)

Examples

			a(6) = lcm(1, tau(1)) * lcm(2, tau(2)) * lcm(3, tau(3)) * lcm(6, tau(6)) = lcm(1, 1) * lcm(2, 2) * lcm(3, 2) * lcm(6, 4) = 1 * 2 * 6 * 12 = 144.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A334782 (Sum_{d|n} lcm(d, tau(d))), A334664 (Product_{d|n} gcd(d, tau(d))).

Cf. A000005 (tau(n)), A009230 (lcm(n, tau(n))).

Programs

Magma

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

Maple

g:= d -> ilcm(d, numtheory:-tau(d)):
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Jun 25 2020

Mathematica

a[n_] := Product[LCM[d, DivisorSigma[0, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], numdiv(d[k]))); \\ Michel Marcus, May 12 2020

Formula

a(p) = 2p for p = odd primes (A065091).