A334470 a(n) = Product_{d|n} (A253139(n) / tau(d)) where A253139(n) = lcm_{d|n} tau(d).

1, 2, 2, 36, 2, 16, 2, 864, 36, 16, 2, 10368, 2, 16, 16, 6480000, 2, 10368, 2, 10368, 16, 16, 2, 11943936, 36, 16, 864, 10368, 2, 4096, 2, 64800000, 16, 16, 16, 2176782336, 2, 16, 16, 11943936, 2, 4096, 2, 10368, 10368, 16, 2, 1343692800000000, 36, 10368, 16

Offset: 1

Author

Jaroslav Krizek, May 01 2020

Keywords

• nonn

Examples

			For n = 6; divisors d of 6: {1, 2, 3, 6}; tau(d): {1, 2, 2, 4}; lcm_{d|6} tau(d) = 4; a(6) = 4/1 * 4/2 * 4/2 * 4/4 = 16.
		

Crossrefs

Cf. A334471 (similar sequence with sigma(d)).

Cf. A000005, A000040, A211776, A253139.

Programs

Magma

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

Mathematica

a[n_] := (LCM @@ (s = DivisorSigma[0, Divisors[n]]))^Length[s] / Times @@ s; Array[a, 51] (* Amiram Eldar, May 02 2020 *)

PARI

a(n) = {my(d=divisors(n), lcmd = lcm(vector(#d, k, numdiv(d[k])))); vecprod(vector(#d, k, lcmd/numdiv(d[k])));} \\ Michel Marcus, May 02 2020

Formula

a(n) = ((lcm_{d|n} tau(d))^tau(n)) / Product_{d|n} tau(d).

a(n) = A253139(n)^A000005(n) / A211776(n).

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

a(n) = 2^(k*2^(k-1)) if n is a product of k distinct primes.