The general term of the prime number sequence and the Smarandache prime function.

Let's consider the function d(i)= as the number of divisors of the positive integer number i. We have found the following expression for this function:

Note: "E(x) is the integer part of x"

We proved this expression in the article: "A functional recurrence to obtain the prime numbers using the Smarandache Prime Function".

We deduce that the following function:

This function is called the Smarandache Prime Function. ( References )

It takes the next values:

Let's consider now number of prime numbers smaller or equal than n.

It is simple to prove that:

Let's have too:

We will see what conditions have to carry .

Therefore we have the following expression for n-th prime number:

If we obtain that only depends on n, this expression will be the general term of the prime numbers sequence, since is in function with G and G does with d(i) that is expressed in function with i too. Therefore the expression only depends on n.

Let's consider

Since from of a certain it will be true that

(1)

If it is not too big, we can prove that the inequality is true for smaller or equal values than .

It is necessary to that:

If we check the inequality:

(2)

We will obtain that:

We can experimentally check this last inequality saying that it checks for a lot of values and the difference tends to increase, which makes to think that it is true for all n.

Therefore if we prove that the (1) and (2) inequalities are true for all n, which seems to be very probable, we will have that the general term of the prime numbers sequence is:

References:

[1]  E. Burton, "Smarandache Prime and Coprime Functions"

[2]  F. Smarandache, "Collected Papers", Vol. II, 200 p., p.137, Kishinev University Press.

Author:

Sebastián Martín Ruiz

Avda de Regla, 43 Chipiona 11550 Cádiz, Spain

E-mail: SMARANDA@teleline.es

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