4 added explanation; edited body
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Mathematica, 66 57 bytes

Length@#1==3&&And@@EqualTo[1]/@#2&@@(FactorInteger@#)&

Defines an anonymous function.

is Transpose.

Explanation

FactorInteger gives a list of pairs of factors and their exponents. E.g. FactorInteger[2250]=={{2,1},{3,2},{5,3}}. This is transposed for ease of use and fed to the function Length@#1==3&&And@@EqualTo[1]/@#2&. The first part, Length@#1==3, checks that there are 3 unique factors, while the second, And@@EqualTo[1]/@#2 checks that all the exponents are 1.

Mathematica, 66 57 bytes

Length@#1==3&&And@@EqualTo[1]/@#2&@@(FactorInteger@#)&

Defines an anonymous function.

is Transpose.

Mathematica, 66 57 bytes

Length@#1==3&&And@@EqualTo[1]/@#2&@@(FactorInteger@#)&

Defines an anonymous function.

is Transpose.

Explanation

FactorInteger gives a list of pairs of factors and their exponents. E.g. FactorInteger[2250]=={{2,1},{3,2},{5,3}}. This is transposed for ease of use and fed to the function Length@#1==3&&And@@EqualTo[1]/@#2&. The first part, Length@#1==3, checks that there are 3 unique factors, while the second, And@@EqualTo[1]/@#2 checks that all the exponents are 1.

3 golfed 9 bytes off; edited body
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Mathematica, 6666 57 bytes

(Length@#1==3&&And@@(EqualTo[1]Length@#1==3&&And@@EqualTo[1]/@#2)@#2&@@(FactorInteger@#)&@@Transpose@FactorInteger@#&&

This is a super weird program structure, but I think it works.

Defines an anonymous function.

is Transpose.

Mathematica, 66 bytes

(Length@#1==3&&And@@(EqualTo[1]/@#2))&@@Transpose@FactorInteger@#&

This is a super weird program structure, but I think it works.

Defines an anonymous function.

Mathematica, 66 57 bytes

Length@#1==3&&And@@EqualTo[1]/@#2&@@(FactorInteger@#)&

Defines an anonymous function.

is Transpose.

2 added 9 characters in body
source | link

Mathematica, 5766 bytes

(Length@#1==3&&And@@(#2==1EqualTo[1]/@#2))&@@Transpose@FactorInteger@#&

This is a super weird program structure, but I think it works.

Defines an anonymous function.

Mathematica, 57 bytes

(Length@#1==3&&And@@(#2==1))&@@Transpose@FactorInteger@#&

This is a super weird program structure, but I think it works.

Defines an anonymous function.

Mathematica, 66 bytes

(Length@#1==3&&And@@(EqualTo[1]/@#2))&@@Transpose@FactorInteger@#&

This is a super weird program structure, but I think it works.

Defines an anonymous function.

1
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