Jelly, 11 bytes

µÆE;¬g/’µ#Ṫ

Try it online!.

Background

Every positive integer k can be factorized uniquely as the product of powers of the first m primes, i.e., k = p₁^α₁⋯p_m^α_m, where α_m > 0.

We have that a^b (b>1) for some positive integer a if and only if b is a divisor of all exponents α_j.

Thus, an integer k > 1 is a perfect power if and only if gcd(α₁, ⋯, α_m) ≠ 1.

How it works

µÆE;¬g/’µ#Ṫ  Main link. No arguments.

µ            Make the chain monadic, setting the left argument to 0.
        µ#   Find the first n integers k, greater or equal to 0, for which the
             preceding chain returns a truthy value.
             In the absence of CLAs, n is read implicitly from STDIN.
 ÆE          Compute the exponents of the prime factorization of k.
   ;¬        Append the logical NOT of k, i.e., 0 if k > 0 and 1 otherwise.
             This maps 1 -> [0] and [0] -> [1].
     g/      Reduce the list of exponents by GCD.
             In particular, we achieved that 1 -> 0 and 0 -> 1.
       ’     Decrement; subtract 1 from the GCD.
             This maps 1 to 0 (falsy) and all other integers to a truthy value.
          Ṫ  Tail; extract the last k.

Jelly, 11 bytes

µÆE;¬g/’µ#Ṫ

Try it online!.

Background

Every positive integer \$k\$ can be factorized uniquely as the product of powers of the first \$m\$ primes, i.e., \$k=p_1^{\alpha_1}\cdots p_m^{\alpha_m}\$, where \$\alpha_m>0\$.

We have that \$a^b\$ (\$b>1\$) for some positive integer \$a\$ if and only if \$b\$ is a divisor of all exponents \$\alpha_j\$.

Thus, an integer \$k > 1\$ is a perfect power if and only if \$\gcd(α_1, ⋯, α_m) ≠ 1\$.

How it works

µÆE;¬g/’µ#Ṫ  Main link. No arguments.

µ            Make the chain monadic, setting the left argument to 0.
        µ#   Find the first n integers k, greater or equal to 0, for which the
             preceding chain returns a truthy value.
             In the absence of CLAs, n is read implicitly from STDIN.
 ÆE          Compute the exponents of the prime factorization of k.
   ;¬        Append the logical NOT of k, i.e., 0 if k > 0 and 1 otherwise.
             This maps 1 -> [0] and [0] -> [1].
     g/      Reduce the list of exponents by GCD.
             In particular, we achieved that 1 -> 0 and 0 -> 1.
       ’     Decrement; subtract 1 from the GCD.
             This maps 1 to 0 (falsy) and all other integers to a truthy value.
          Ṫ  Tail; extract the last k.

Jelly, 11 bytes

µÆE;¬g/’µ#Ṫ

Try it online!.

Background

Every positive integer k can be factorized uniquely as the product of powers of the first m primes, i.e., k = p₁^α₁⋯p_m^α_n, where α_n > 0.

We have that a^b (b>1) for some positive integer a if and only if b is a divisor of all exponents α_j.

Thus, an integer k > 1 is a perfect power if and only if gcd(α₁, ⋯, α_m) ≠ 1.

How it works

µÆE;¬g/’µ#Ṫ  Main link. No arguments.

µ            Make the chain monadic, setting the left argument to 0.
        µ#   Find the first n integers k, greater or equal to 0, for which the
             preceding chain returns a truthy value.
             In the absence of CLAs, n is read implicitly from STDIN.
 ÆE          Compute the exponents of the prime factorization of k.
   ;¬        Append the logical NOT of k, i.e., 0 if k > 0 and 1 otherwise.
             This maps 1 -> [0] and [0] -> [1].
     g/      Reduce the list of exponents by GCD.
             In particular, we achieved that 1 -> 0 and 0 -> 1.
       ’     Decrement; subtract 1 from the GCD.
             This maps 1 to 0 (falsy) and all other integers to a truthy value.
          Ṫ  Tail; extract the last k.

Jelly, 11 bytes

µÆE;¬g/’µ#Ṫ

Try it online!.

Background

Every positive integer k can be factorized uniquely as the product of powers of the first m primes, i.e., k = p₁^α₁⋯p_m^α_m, where α_m > 0.

We have that a^b (b>1) for some positive integer a if and only if b is a divisor of all exponents α_j.

Thus, an integer k > 1 is a perfect power if and only if gcd(α₁, ⋯, α_m) ≠ 1.

How it works

µÆE;¬g/’µ#Ṫ  Main link. No arguments.

µ            Make the chain monadic, setting the left argument to 0.
        µ#   Find the first n integers k, greater or equal to 0, for which the
             preceding chain returns a truthy value.
             In the absence of CLAs, n is read implicitly from STDIN.
 ÆE          Compute the exponents of the prime factorization of k.
   ;¬        Append the logical NOT of k, i.e., 0 if k > 0 and 1 otherwise.
             This maps 1 -> [0] and [0] -> [1].
     g/      Reduce the list of exponents by GCD.
             In particular, we achieved that 1 -> 0 and 0 -> 1.
       ’     Decrement; subtract 1 from the GCD.
             This maps 1 to 0 (falsy) and all other integers to a truthy value.
          Ṫ  Tail; extract the last k.

Jelly, 11 bytes

µÆE;¬g/’µ#Ṫ

Try it online!.

How it works

µÆE;¬g/’µ#Ṫ  Main link. No arguments.

µ            Make the chain monadic, setting the left argument to 0.
        µ#   Find the first n integers k, greater or equal to 0, for which the
             preceding chain returns a truthy value.
             In the absence of CLAs, n is read implicitly from STDIN.
 ÆE          Compute the exponents of the prime factorization of k.
   ;¬        Append the logical NOT of k, i.e., 0 if k > 0 and 1 otherwise.
             This maps 1 -> [0] and [0] -> [1].
     g/      Reduce the list of exponents by GCD.
             In particular, we achieved that 1 -> 0 and 0 -> 1.
       ’     Decrement; subtract 1 from the GCD.
             This maps 1 to 0 (falsy) and all other integers to a truthy value.
          Ṫ  Tail; extract the last k.

Jelly, 11 bytes

µÆE;¬g/’µ#Ṫ

Try it online!.

Background

Every positive integer k can be factorized uniquely as the product of powers of the first m primes, i.e., k = p₁^α₁⋯p_m^α_n, where α_n > 0.

We have that a^b (b>1) for some positive integer a if and only if b is a divisor of all exponents α_j.

Thus, an integer k > 1 is a perfect power if and only if gcd(α₁, ⋯, α_m) ≠ 1.

How it works

µÆE;¬g/’µ#Ṫ  Main link. No arguments.

µ            Make the chain monadic, setting the left argument to 0.
        µ#   Find the first n integers k, greater or equal to 0, for which the
             preceding chain returns a truthy value.
             In the absence of CLAs, n is read implicitly from STDIN.
 ÆE          Compute the exponents of the prime factorization of k.
   ;¬        Append the logical NOT of k, i.e., 0 if k > 0 and 1 otherwise.
             This maps 1 -> [0] and [0] -> [1].
     g/      Reduce the list of exponents by GCD.
             In particular, we achieved that 1 -> 0 and 0 -> 1.
       ’     Decrement; subtract 1 from the GCD.
             This maps 1 to 0 (falsy) and all other integers to a truthy value.
          Ṫ  Tail; extract the last k.