A087320 Smallest perfect power (at least a square) that is a multiple of n.

1, 4, 9, 4, 25, 36, 49, 8, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 144, 25, 676, 27, 196, 841, 900, 961, 32, 1089, 1156, 1225, 36, 1369, 1444, 1521, 400, 1681, 1764, 1849, 484, 225, 2116, 2209, 144, 49, 100, 2601, 676, 2809, 216, 3025

Offset: 1

Author

Amarnath Murthy, Sep 03 2003

Keywords

• easy
• nonn

Comments

The prime signature of a(n) is not determined by the prime signature of n. For example, a(2^3*3^5) = 2^3*3^6, but a(2^3*5^5) = 2^5*5^5. - David Wasserman, May 03 2005

Likewise, this sequence is not multiplicative. The smallest exception is a(24) = 144 > 72 = a(3)*a(8). - Franklin T. Adams-Watters, Oct 18 2011

Examples

			a(54) = 216 = 6^3. 54 is the least n such that a(n)/n does not divide A007947(n).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001597, A007947, A087321, A160400, A197863.

Programs

Mathematica

a[n_] := n * SelectFirst[Range[n], GCD @@ FactorInteger[n*#][[;; , 2]] > 1 &]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

Formula

a(n)/n <= A007947(n) (the squarefree kernel of n).

a(p^k) = p^k, a(k) = A007947(k)^2 for cubefree k. Furthermore, the upper bound on a(n)/n can be tightened to A007913(n). - Charlie Neder, Dec 26 2018

From Amiram Eldar, Jul 09 2022: (Start)

a(n) = n iff n is in A001597.

a(n) = n * A160400(n). (End)

Extensions

Edited and extended by David Wasserman, May 03 2005