Triplet in Arithmetic Progression with equal Sigma

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Theorem

The smallest triple of integers in arithmetic progression which have the same $\sigma$ (sigma) value is:

$\sigma \left({267}\right) = \sigma \left({295}\right) = \sigma \left({323}\right) = 360$


Proof

We have that:

\(\displaystyle 295 - 267\) \(=\) \(\displaystyle 28\)
\(\displaystyle 323 - 295\) \(=\) \(\displaystyle 28\)

demonstrating that $267, 295, 323$ are in arithmetic progression with common difference $28$.


Then:

\(\displaystyle \sigma \left({267}\right)\) \(=\) \(\displaystyle 360\) Sigma of 267
\(\displaystyle \sigma \left({295}\right)\) \(=\) \(\displaystyle 360\) Sigma of 295
\(\displaystyle \sigma \left({323}\right)\) \(=\) \(\displaystyle 360\) Sigma of 323



Sources