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Erdős problem 906

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Is there an entire non-zero function f:C→Cf:\mathbb{C}\to \mathbb{C} such that, for any infinite sequence n1<n2<⋯n_1<n_2<\cdots, the set {z:f(nk)(z)=0 for some k≥1}\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\} is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.

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Is there an entire non-zero function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set $\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}$ is everywhere dense? The literal question is trivial for polynomials, so the claims address the transcendental entire case, in the affirmative.

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