## factorial and arithmetic functions

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• 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

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