Problems & Puzzles: Puzzles

Puzzle 1183  F(n)=(Floor(n!/E) + Mod(n, 2))/n Sebastián Martín Ruiz, sent the following puzzle: Let F(n)=(Floor(n!/E) + Mod(n, 2))/n where E=2,712828... and Mod(n,2)=0 or 1 if n is even or odd Q) Prove that F(n) is a composite integer number except for n=4 which is F(4)=2 or find a counterexample.

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From Aug 3-9, 2024, contributions came from Michael Branicky, Emmanuel Vantieghem

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Michael wrote:

F(n) = A000166(n) in the OEIS.   

F(4) = 2 is noted as the only prime in a comment there (Berman, Nov 08 2008).
 

This can also be seen in the first comment, namely, Euler proved the recurrence

F(n) = (n-1)*(F(n-1) + F(n-2)), with F(0) = 1 and F(1) = 0 (*)
 

From (*), it is easy to see that F(n) is divisible by n-1 for n > 2. #

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Emmanuel wrote:

This is what I found about Puzzle 1183 :

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