A334471 -id:A334471 - cp's OEIS frontend

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

Jaroslav Krizek, May 01 2020

nonn

			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.

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

Cf. A000005, A000040, A211776, A253139.

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

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

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

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.

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

Original entry on oeis.org

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

Jaroslav Krizek, May 03 2020

nonn

			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.

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

Cf. A000005, A000040, A007955, A266265.

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

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

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

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

cp's OEIS Frontend

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

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