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How to count

Dear Mathematica Gurus!

I have seriously problem with counting distances between smallest square 
bigger than n! (for big n) e.g.

aa = {}; Do[k = 1 + Floor[Sqrt[n!]]; kk = k^2 - n!; Print[kk];
  kkk = N[Log[kk], 100]; Print[kkk];
  AppendTo[aa, kkk], {n, 1, 10000000, 1000000}]; aa

Who have idea how to count this on Mathematica?

Best wishes

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