A304328 -id:A304328 - cp's OEIS frontend

A304326 Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.

0, 1, 1, 1, 1, 3, 1, 0, 1, 3, 1, 3, 1, 3, 3, 0, 1, 3, 1, 3, 3, 3, 1, 2, 1, 3, 0, 3, 1, 7, 1, 0, 3, 3, 3, 3, 1, 3, 3, 2, 1, 7, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 7, 1, 3, 3, 0, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 2, 0, 3, 1, 7, 3, 3, 3, 2, 1

Offset: 1

Gus Wiseman, May 10 2018

nonn

			The a(180) = 7 ways are (6*30), (12*15), (18*10), (30*6), (60*3), (90*2), (180*1).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Positions of zeros are A246549. Range appears to be A075427.

Cf. A000961, A001055, A001597, A001694, A005117, A007916, A034444, A091050, A183096, A303386, A303707, A304327, A304328.

radQ[n_]:=And[n>1,GCD@@FactorInteger[n][[All,2]]===1];
Table[Length[Select[Divisors[n],radQ[#]&&SquareFreeQ[n/#]&]],{n,100}]

a(n)={sumdiv(n, d, d<>1 && !ispower(d) && issquarefree(n/d))} \\ Andrew Howroyd, Aug 26 2018

A304327 Number of ways to write n as a product of a perfect power and a squarefree number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Gus Wiseman, May 10 2018

nonn

First term greater than 2 is a(746496) = 3.

			The a(746496) = 3 ways are 12^5*3, 72^3*2, 864^2*1.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000961, A001055, A001597, A005117, A007916, A034444, A091050, A183096, A203025, A246549, A303386, A304326, A304328.

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

A304327(n) = sumdiv(n,d,issquarefree(n/d)*((1==d)||ispower(d))); \\ Antti Karttunen, Jul 29 2018

More terms from Antti Karttunen, Jul 29 2018

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

A304326 Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A304327 Number of ways to write n as a product of a perfect power and a squarefree number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI