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factorial and arithmetic functions

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  • Sebastian Martin
    Hello All These formulas with factorials and arithetic functions are already kown?: If n is prime: DivisorSigmak(n!)/DivisorSigmak((n-1)!)=n^k + 1 If n is
    Message 1 of 1 , Feb 21, 2006
      Hello All

      These formulas with factorials and arithetic functions

      are already kown?:

      If n is prime:

      DivisorSigmak(n!)/DivisorSigmak((n-1)!)=n^k + 1

      If n is composite:

      DivisorSigmak(n!)/DivisorSigmak((n-1)!) aprox= n^k

      where:

      DivisorSigmak= sum of the kth powers of divisors of n

      aprox= aproximately equal

      And for the Euler Totient function we have other
      formulas:

      If n is composite:

      Eulerphi(n!)=nEulerphi((n-1)!)

      If n is prime:

      EulerPhi(n!)=(n-1)EulerPhi((n-1))!

      We can deduce the following (rule mnemonics?) (In
      spanish REGLA MNEMOTECNICA) for the evaluation of
      EulerPhi(n!):

      Example: Phi(120)=Phi(5*4*3*2*1)=4*4*2*1*1=32
      Only we must change the primes in the factorial for
      the primes minus one.

      Another example:
      Phi(720)=Phi(6*5*4*3*2*1)=6*4*4*2*1*1=192

      CAN ANYONE PROVE ALL THIS?

      CAN ANYONE TO EXTEND THIS FOR OTHERS ATRITHMETICS
      FUNCTIONS?

      Sincerely

      Sebastian Martin Ruiz

      http://perso.wanadoo.es/smaranda/








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