Nonnegative numbers n for which n^2 - n - 56 is not squarefree.

0, 1, 2, 3, 4, 5, 8, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 26, 28, 29, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 71, 72, 73, 74, 76, 77, 78, 80, 81, 83, 84, 85, 86, 88, 89, 91

1

For numbers q between -100 and 100, q = -56 appears to produce the least number of squarefree numbers for the quadratic n^2 - n - q.

T. D. Noe, Plot of 1000 terms

T. D. Noe, Table of 1000 terms

(Mma) Select[Range[0, 200], ! SquareFreeQ[#^2 - # - 56] &]

nonn

T. D. Noe, Jul 22 2014