Does there exist a holomorphic function which takes given values on the positive integers? Inspired of course by What's a natural candidate for an analytic function that interpolates the tower function?
I am minded to ask what looks to me like a more natural question: given a sequence $a_1,a_2,a_3,\ldots$ of complex numbers, is there always a holomorphic function $f$ defined on the entire complex plane, with $f(n)=a_n$ for $n=1,2,3,\ldots$? No idea what the answer is myself, but wouldn't surprise me if it were well-known and even easy.
 A: Probably well-known. Easy? I'd venture to guess that an expression like $$\sum_{n=1}^\infty b_n\frac{e^{c_n(z-n)}}{(n-1)!}\prod_{k=1}^{n-1}(z-k)$$ can be made to work. You'll have to pick the $b_n$ successively to make the $N$'th partial sum equal to $a_N$, and real constants $c_n$ large enough to obtain uniform convergence to the left of any fixed vertical line. E.g., so that the $n$'th term has absolute value less than $2^{-n}$ when $\operatorname{Re} z<n/2$. 
A: This is Exercise 6, Page 26, of Knopp's Problem Book in the Theory of Functions, Volume 2: For any sequence of complex numbers $z_n$ with no finite limit point, and for any sequence of complex numbers $w_n$, there is an entire function mapping $z_n$ to $w_n$. The proof goes like this: Use the Weierstrass Factor Theorem to construct a function $W$ with simple zeros at the $z_n$. Use the Mittag-Leffler theorem to construct a function $M$ with simple poles at the $z_n$ with residues $\frac{w_n}{W'(z_n)}$. Then the function $W\cdot M$ does the job.
