Consecutive integers which are not squarefree

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8 and 9 are the only consecutive integers which are perfect powers (Catalan's conjecture, proved in 2002 by Preda Mihăilescu). Thus for 3 or more consecutive integers which are not squarefree there is one or more Achilles numbers between any perfect powers, Achilles numbers being defined as nonsquarefree numbers which are not perfect powers.

First run of (at least) n consecutive integers which are not squarefree

A045882 Smallest term of first run of (at least) $\scriptstyle n\,$ consecutive integers which are not squarefree.

{4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, ...}

First run of exactly n consecutive integers which are not squarefree

A051681 Smallest term of first run of exactly $\scriptstyle n\,$ consecutive integers which are not squarefree.

{4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 262315467, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, ...}

Sequences

A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

A049535 Starts for strings of exactly 6 consecutive non-squarefree numbers.

A077640 Smallest term of a run of at least 7 consecutive integers which are not squarefree.

A077647 Smallest term of a run of at least 8 consecutive integers which are not squarefree.

A078143 Smallest term of a run of at least 9 consecutive integers which are not squarefree.

See also

• Consecutive integers which are perfect powers