login
a(n) = Product_{d|n} (pod(n)/pod(d)) where pod(n) = A007955(n), the product of divisors of n.
0

%I #19 Sep 08 2022 08:46:25

%S 1,2,3,32,5,7776,7,16384,243,100000,11,8916100448256,13,537824,759375,

%T 1073741824,17,1156831381426176,19,4096000000000000,4084101,5153632,

%U 23,2315513501476187716057433112576,3125,11881376,4782969,232218265089212416,29

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

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

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

%F a(n) = n^c(n) where c(n) only depends on the prime signature of n. - _David A. Corneth_, May 05 2020

%e 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.

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

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

%o (PARI) pod(n) = vecprod(divisors(n));

%o a(n) = my(d=divisors(n), podn = pod(n)); prod(k=1, #d, podn/pod(d[k])); \\ _Michel Marcus_, May 03-11 2020

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

%Y Cf. A000005, A000040, A007955, A266265.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, May 03 2020