On the Srnarandache Irrationality Conjecture

Here is an immediate proof in the following cases: 1. a(n) = n: 2. a(n) = d(n) =number of divisors of n; 3. a(n) = w(n) =number of distinct prime divisors of n: 4. a(n) = D(n) =number of total prime divisors of n (that is. counted with repetitions): 5. a(n) = dJ(n) =the Euler function of n: 6. a( n) = cr( n) =the sum of the divisors of n; 7. a(n) = Pn =the nth prime: 8. a(n) = 71(n) =the number of primes smaller than n: 9. a(n) = S(n) =the Smarandache function of n; 10. a(n) = n!; 11. a(n) = an, where a is any fixed positive integer coprime to 10 and larger than 1; 12. a(n) =any fixed non-constant polynomial in one of the above;