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 Puzzle 352. φ(n!) Sebastián Martín Ruiz found empirically that: φ(n!) = n.φ((n-1)!) or (n-1).φ((n-1)!), if n is composite or prime, respectively...................(1) As a matter of fact I have proved, by deduction, the following expression: φ(n!) = Π(i-k(i)); i=1 to n; k(i)=0 if i is composite or k(i)=1 is i is prime; k(1)=0.........................(2) from which are generated the expressions discovered by SMR. From (2), here is an illustration:      10!=10.9.8.7.6.5.4.3.2.1 φ(10!)=10.9.8.6.6.4.4.2.1.1   Question: Can you send your own proof of (2)?

Contributions came from Joseph L. Pe, Farideh Firoozbakht, Rudolph Knjzek, Salvatore Ingala, Dan Dima.

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Only Dan Dima proved directly (2). His solution will be posted soon. The other puzzlers first proved (1) and then (2) by induction.

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