Triplet in Arithmetic Sequence with equal Divisor Sum

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Theorem

The smallest triple of integers in arithmetic sequence which have the same divisor sum is:

$\map {\sigma_1} {267} = \map {\sigma_1} {295} = \map {\sigma_1} {323} = 360$


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:

\(\ds \map {\sigma_1} {267}\) \(=\) \(\ds 360\) $\sigma_1$ of $267$
\(\ds \map {\sigma_1} {295}\) \(=\) \(\ds 360\) $\sigma_1$ of $295$
\(\ds \map {\sigma_1} {323}\) \(=\) \(\ds 360\) $\sigma_1$ of $323$



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Sources

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