Fibonacci + Factorials = Fibatorials!

We define:

  • \(F(n)\) denote the \(n\)-th Fibonacci Number.
  • \(\Psi (n)\) denote the sum of digits of \(n\).
  • \(\phi (n)\) denote the sum of digits of \(n!\).

Find the sum of all integers between 1 and 1000 inclusive such that \(\phi (n) = \phi (\Psi (F(n)))\).

Note: \(F_0 = 0, F_1=1 \).

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