Squareful
A number is squareful, also called nonsquarefree, if it contains at least one square in its prime factorization. The first few are 4, 8, 9, 12, 16, 18, 20, 24, 25, ... (OEIS A013929). The greatest multiple prime factors for the squareful integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 3, ... (OEIS A046028). The least multiple prime factors for squareful integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, ... (OEIS A046027).
No squareful Fibonacci numbers 
are known with
prime.
The sequence of smallest terms in the first run of (at least) 
consecutive squareful
integers for 
, 2, ... is 4, 8, 48, 242, 844, 22020,
217070, ... (OEIS A045882).
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2x^2 - 3xy + 4y^2 + 6x - 3y - 4 = 0