# Sequence of 4 Consecutive Square-Free Triplets

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## Theorem

The following sets of $4$ consecutive triplets of integers, with one integer between each triplet, are square-free:

- $29, 30, 31; 33, 34, 35; 37, 38, 39; 41, 42, 43$

- $101, 102, 103; 105, 106, 107; 109, 110, 111; 113, 114, 115$

## Proof

Note that $32, 36, 40$ and $104, 108, 112$ are all divisible by $4 = 2^2$, so are by definition not square-free.

Then inspecting each number in turn:

\(\ds 29\) | \(\) | \(\ds \) | is prime | |||||||||||

\(\ds 30\) | \(=\) | \(\ds 2 \times 3 \times 5\) | and so is square-free | |||||||||||

\(\ds 31\) | \(\) | \(\ds \) | is prime |

\(\ds 33\) | \(=\) | \(\ds 3 \times 11\) | and so is square-free | |||||||||||

\(\ds 34\) | \(=\) | \(\ds 2 \times 17\) | and so is square-free | |||||||||||

\(\ds 35\) | \(=\) | \(\ds 5 \times 7\) | and so is square-free |

\(\ds 37\) | \(\) | \(\ds \) | is prime | |||||||||||

\(\ds 38\) | \(=\) | \(\ds 2 \times 19\) | and so is square-free | |||||||||||

\(\ds 39\) | \(=\) | \(\ds 3 \times 13\) | and so is square-free |

\(\ds 41\) | \(\) | \(\ds \) | is prime | |||||||||||

\(\ds 42\) | \(=\) | \(\ds 2 \times 3 \times 7\) | and so is square-free | |||||||||||

\(\ds 43\) | \(\) | \(\ds \) | is prime |

\(\ds 101\) | \(\) | \(\ds \) | is prime | |||||||||||

\(\ds 102\) | \(=\) | \(\ds 2 \times 3 \times 17\) | and so is square-free | |||||||||||

\(\ds 103\) | \(\) | \(\ds \) | is prime |

\(\ds 105\) | \(=\) | \(\ds 3 \times 5 \times 7\) | and so is square-free | |||||||||||

\(\ds 106\) | \(=\) | \(\ds 2 \times 53\) | and so is square-free | |||||||||||

\(\ds 107\) | \(\) | \(\ds \) | is prime |

\(\ds 109\) | \(\) | \(\ds \) | is prime | |||||||||||

\(\ds 110\) | \(=\) | \(\ds 2 \times 5 \times 11\) | and so is square-free | |||||||||||

\(\ds 111\) | \(=\) | \(\ds 3 \times 37\) | and so is square-free |

\(\ds 113\) | \(\) | \(\ds \) | is prime | |||||||||||

\(\ds 114\) | \(=\) | \(\ds 2 \times 3 \times 19\) | and so is square-free | |||||||||||

\(\ds 115\) | \(=\) | \(\ds 5 \times 27\) | and so is square-free |

$\blacksquare$

## Also see

This sequence is A007675 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $29$