Erdős Problem #360
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
\(f(n)\): the least number of classes in a partition of \(\{1,\ldots,n-1\}\) such that \(n\) is not a sum of distinct elements of one class. A class contains \(n\) among its subset sums exactly when some finite subset of the class sums to \(n\). For \(n \geq 2\) the partition into singletons is admissible (each element is less than \(n\)), so the set is non-empty and `sInf` is its minimum. For \(n \leq 1\) the set \(\{1,\ldots,n-1\}\) is empty, the empty partition is admissible, and \(f(n) = 0\).
The answer: \(\frac{n^{1/3}(n/\phi(n))}{(\log n)^{1/3}(\log\log n)^{2/3}}\).
Attempts
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