# 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

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $267$