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Please help me with this problem

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 \(n\)?

Note by Keshav Tiwari
5 months, 2 weeks ago

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@Calvin Lin Keshav Tiwari · 5 months, 1 week ago

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@Keshav Tiwari Yes there is a way.

What have you tried? Calvin Lin Staff · 5 months, 1 week ago

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@Calvin Lin 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. Keshav Tiwari · 5 months, 1 week ago

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@Keshav Tiwari Try smoothing an inequality. Calvin Lin Staff · 5 months, 1 week ago

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