Derive max distinct frequencies for n items

Q: Derive max distinct frequencies for n items

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Question

Maximum Number of Distinct Frequency Counts in an Array

Context

You are given an array of length n ≥ 1 whose elements are arbitrary integers (values may repeat). For each distinct value, compute its frequency (number of occurrences). Among these frequencies, some values may coincide. We ask:

What is the maximum possible number of distinct frequency values that can appear?
Give a tight expression in terms of n, prove it is optimal, and present constructions that achieve the bound.

Task

Define the function m_max(n): the maximum number of distinct frequency counts achievable by any length-n array.
Derive a closed-form expression for m_max(n) in terms of n.
Prove optimality (upper bound and matching construction).
Provide small illustrative examples.

Derive max distinct frequencies for n items

Maximum Number of Distinct Frequency Counts in an Array

Context

Task

Solution (Locked)

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