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

1,2

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

Links

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

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)

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A345304 A078615 A197863 * A087321 A053143 A068238

Adjacent sequences: A087317 A087318 A087319 * A087321 A087322 A087323

Keyword

easy,nonn

Author

Amarnath Murthy, Sep 03 2003

Extensions

Edited and extended by David Wasserman, May 03 2005

Status

approved