login
A087320
Smallest perfect power (at least a square) that is a multiple of n.
11
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
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
KEYWORD
easy,nonn
AUTHOR
Amarnath Murthy, Sep 03 2003
EXTENSIONS
Edited and extended by David Wasserman, May 03 2005
STATUS
approved