Are Aadhaar Numbers Generated Algorithmically? - Yes.

Aadhaar Number Allocaiton or Generation

Are Aadhaar Numbers Generated Algorithmically?

The logic behind the PAN Card is well explained in the link below, 

https://en.wikipedia.org/wiki/Permanent_account_number

The following link explains the 12 digit aadhaar number, 

https://www.quora.com/What-is-the-structure-of-ones-Aadhar-Card-UID-number

Based on the Biometrics (facial Photo, Iris and 10 FPs) information of an individual, a 12 digit unique random number generated after de-duplication and stages of SEDA.