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Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).
4

%I #7 Sep 28 2024 07:33:16

%S 1,1,1,1,2,1,1,1,1,4,1,1,2,3,1,1,1,2,1,1,1,1,1,1,2,1,1,2,4,1,1,1,1,2,

%T 1,1,1,1,2,1,1,2,1,1,1,2,1,1,1,1,1,1,3,1,1,1,2,1,1,1,4,1,1,1,2,1,1,3,

%U 1,2,2,1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,2

%N Run-lengths of first differences (A078147) of nonsquarefree numbers (A013929).

%e The sequence of nonsquarefree numbers (A013929) is:

%e 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...

%e with first differences (A078147):

%e 4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, ...

%e with runs:

%e (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...

%e with lengths (A376267):

%e 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...

%t Length/@Split[Differences[Select[Range[1000], !SquareFreeQ[#]&]]]//Most

%Y For prime instead of nonsquarefree numbers we have A333254.

%Y For run-sums instead of run-lengths we have A376264.

%Y For squarefree instead of nonsquarefree we have A376306.

%Y For prime-powers instead of nonsquarefree numbers we have A376309.

%Y For compression instead of run-lengths we have A376312.

%Y A000040 lists the prime numbers, differences A001223.

%Y A000961 and A246655 list prime-powers, differences A057820.

%Y A005117 lists squarefree numbers, differences A076259 (ones A375927).

%Y A013929 lists nonsquarefree numbers, differences A078147.

%Y Cf. A007674, A053797, A053806, A072284, A112925, A120992, A373198, A375707, A376305, A376307, A376311.

%K nonn

%O 1,5

%A _Gus Wiseman_, Sep 27 2024