A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 64, 1, 1, 1, 16, 1, 64, 1, 1024, 1, 1, 1, 729, 32, 1, 27, 16384, 1, 1, 1, 25, 1, 1, 1, 4096, 1, 1, 1, 59049, 1, 1, 1, 4194304, 32768, 1, 1, 4096, 128, 1024, 1, 67108864, 1, 729, 1, 4782969, 1, 1, 1, 1073741824, 1, 1, 2097152, 36, 1, 1, 1, 17179869184, 1, 1, 1, 46656, 1, 1, 32768

Offset: 1

Author

Ilya Gutkovskiy, Apr 20 2018

Keywords

• nonn

Comments

This is different from A008477, which is Product_j k_j^p_j. - N. J. A. Sloane, May 01 2021

Examples

			a(36) = a(2^2 * 3^2) = (2*2)^(2*3) = 4^6 = 4096.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..4096

Crossrefs

Cf. A000026, A000142, A005117 (indices of ones), A005361, A007947, A008477, A034386, A039696, A135291, A285769, A303277.

Programs

Mathematica

Table[Times@@Transpose[FactorInteger[n]][[2]]^Last[Select[Divisors[n], SquareFreeQ]], {n, 75}]

PARI

a(n) = my(f=factor(n)); factorback(f[, 2])^factorback(f[, 1]); \\ Michel Marcus, Apr 21 2018

Formula

a(n) = tau(n/rad(n))^rad(n) = A005361(n)^A007947(n).

a(p^k) = k^p where p is a prime.

a(A000142(k)) = A135291(k)^A034386(k).

Extensions

Definition clarified by N. J. A. Sloane, May 01 2021