A160400 -id:A160400 - cp's OEIS frontend

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

Amarnath Murthy, Sep 03 2003

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

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

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

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

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

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)

Edited and extended by David Wasserman, May 03 2005

A304328 a(n) = n/(largest perfect power divisor of n).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 10 2018

nonn

Not all terms are squarefree numbers; for example, a(500) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001055, A001222, A001597, A001694, A005117, A007716, A007916, A052485, A052486, A059404, A066636, A091050, A160400, A203025, A294068, A303707, A304327, A304339.

Table[n/Last[Select[Divisors[n],#===1||GCD@@FactorInteger[#][[All,2]]>1&]],{n,100}]

a(n)={my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); n/m} \\ Andrew Howroyd, Aug 26 2018

a(n) * A203025(n) = n.

A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 2

Offset: 1

Gus Wiseman, May 11 2018

nonn

All terms are squarefree numbers. First differs from A304328 at a(500) = 1, A304328(500) = 4.

			f maps 500 -> 4 -> 1 -> 1, so a(500) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001597, A001694, A005117, A007916, A052485, A052486, A059404, A091050, A160400, A203025, A294068, A303707, A304327, A304328.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
op[n_]:=n/Last[Select[Divisors[n],!radQ[#]&]];
Table[FixedPoint[op,n],{n,200}]

a(n)={while(1, my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); if(m==1, return(n)); n/=m)} \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

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

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A304328 a(n) = n/(largest perfect power divisor of n).

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A304339 Fixed point of f starting with n, where f(x) = x/(largest perfect power divisor of x).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI