Hello everyone!

I came across a strange question today. Consider a number say 10, we will try to express it as a sum of positive integers say

\[ 10 = \underbrace{1+1+1+\cdots + 1}_{\text{ten 1's}} = 2 + 4 + 4 = 2 + 3 + 5 = \cdots \]

We find the maximum possible lowest common multiple of these numbers and call it \(S_{10} \), so \(S_{10} = 2\times3\times5=30\).

Other examples are \(S_{7} = 3\times4=12, S_{8} = 3\times5=15\).

Is there a way to find \(S_{n}\) for all positive integers \(n\)?

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TopNewest@Calvin Lin

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Yes there is a way.

What have you tried?

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Not much ,I did it by hit and trial for small numbers( as most questions asked only about them) .I have no idea on how to proceed with large numbers.

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smoothing an inequality.

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