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It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth function with this property.But what if we require the function to be real analytic ? Does there exists an analytic function $P$ that has the property $P(n)=p_n$ and its taylor series has integer coefficients?

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Taylor series with nontrivial integral coefficients usually have a small radius of convergence. Do you actually mean that the nth coefficient is (a_n)/n! for integers a_n? Or do you want the series expansion around every integer to have integer coefficients? Gerhard "Scratching His Head On This" Paseman, 2012.03.02 – Gerhard Paseman Mar 2 2012 at 10:56
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 Specifically, a Taylor series with integer coefficients, infinitely many of which are nonzero, has radius of convergence at most $1$. – Emil Jeřábek Mar 2 2012 at 11:42
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 (In case it’s not obvious, it’s easy to find an entire function which does the job if we allow the coefficients to be real.) – Emil Jeřábek Mar 2 2012 at 12:06
@Paseman I mean a_n is integer, – nemesiso Mar 2 2012 at 13:39

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