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Husk, 9 bytes

fo§¦ṁ\LḊN

Try it online! (header outputs the first few elements to avoid timing-out)

 N # from the infinite list of integers
fo # output those for which
 ṁ\ # the sum of the reciprocals of their divisors
 §¦ # exactly divides
 LḊ # the length (number) of their divisors
```