A334489 a(n) = Product_{d|n} (pod(n)/pod(d)) where pod(n) = A007955(n), the product of divisors of n.

1, 2, 3, 32, 5, 7776, 7, 16384, 243, 100000, 11, 8916100448256, 13, 537824, 759375, 1073741824, 17, 1156831381426176, 19, 4096000000000000, 4084101, 5153632, 23, 2315513501476187716057433112576, 3125, 11881376, 4782969, 232218265089212416, 29

Offset

1,2

Links

Table of n, a(n) for n=1..29.

Formula

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

a(n) = ((lcm_{d|n} pod(d))^tau(n)) / Product_{d|n} (pod(d)) = A007955(n)^A000005(n)/A266265(n).

a(n) = n^c(n) where c(n) only depends on the prime signature of n. - David A. Corneth, May 05 2020

Example

For n = 6; divisors d of 6: {1, 2, 3, 6}; pod(d): {1, 2, 3, 36}; lcm_{d|6} pod(d) = pod(6) = 36; a(6) = 36/1 * 36/2 * 36/3 * 36/36 = 7776.

Mathematica

pod[n_] := Times @@ Divisors[n]; a[n_] := pod[n]^Length[(d = Divisors[n])]/Times @@ (pod /@ d); Array[a, 30] (* Amiram Eldar, May 03 2020 *)

Prog

(Magma) [&*[ LCM([&*Divisors(d): d in Divisors(n)]) / &*Divisors(d): d in Divisors(n)]: n in [1..100]]
(PARI) pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n), podn = pod(n)); prod(k=1, #d, podn/pod(d[k])); \\ Michel Marcus, May 03-11 2020

Crossrefs

Cf. Similar sequences for functions lcm_{d|n} tau(d) and lcm_{d|n} sigma(d): A334470, A334471.

Cf. A000005, A000040, A007955, A266265.

Sequence in context: A128030 A239593 A214658 * A217761 A004843 A173353

Adjacent sequences: A334486 A334487 A334488 * A334490 A334491 A334492

Keyword

nonn

Author

Jaroslav Krizek, May 03 2020

Status

approved