The sequence A000120 is a familiar example of such a function. For other examples, see the answers to this question: Existence of a sequence that has every element of $\mathbb N$ infinite number of times
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1I'm not aware of any term for such a thing. – Qiaochu Yuan Sep 10 '22 at 01:18
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1Perhaps an "infinity-to-one function". E.g. https://math.stackexchange.com/questions/1599921/countable-infinity-to-one-function – paw88789 Sep 10 '22 at 01:28
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"infinity-to-one" is what I was thinking of. – firstname gklsodascb Sep 10 '22 at 02:14