Jelly, (削除) 14 (削除ここまで) 13 bytes

‘‘ÆE’Pȧg/Ʋ’Ɗ¿

Try it online!

Did not look at the linked Jelly answer, but that byte count does look familiar... until I remembered how I was originally checking the GCD anyhow

edit: they're not even remotely alike what the hell

Jelly, (削除) 14 (削除ここまで) 13 bytes

‘‘ÆE’Pȧg/Ʋ’Ɗ¿

Try it online!

Did not look at the linked Jelly answer, but that byte count does look familiar... until I remembered how I was originally checking the GCD anyhow

edit: they're not even remotely alike what the hell

‘ Starting at 1 more than the input,
 ‘ Ɗ¿ increment while:
 ÆE Ʋ for the exponents of the prime factorization,
 P the product of
 ’ each minus 1
 ȧ logical-AND
 g/ their greatest common denominator
 ’ is not equal to 1.

Jelly, 14 bytes

‘‘ÆE’Pȧg/Ʋ−1Ʋ¿

Try it online!

Did not look at the linked Jelly answer, but that byte count does look familiar...

edit: they're not even remotely alike what the hell

Jelly, (削除) 14 (削除ここまで) 13 bytes

‘‘ÆE’Pȧg/Ʋ’Ɗ¿

Try it online!

Did not look at the linked Jelly answer, but that byte count does look familiar... until I remembered how I was originally checking the GCD anyhow

edit: they're not even remotely alike what the hell