Triplet in Arithmetic Sequence with equal Sigma

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Theorem

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

$\map \sigma {267} = \map \sigma {295} = \map \sigma {323} = 360$


Proof

We have that:

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

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


Then:

\(\ds \map \sigma {267}\) \(=\) \(\ds 360\) $\sigma$ of $267$
\(\ds \map \sigma {295}\) \(=\) \(\ds 360\) $\sigma$ of $295$
\(\ds \map \sigma {323}\) \(=\) \(\ds 360\) $\sigma$ of $323$



Sources