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Run-compression of first differences (A078147) of nonsquarefree numbers (A013929).
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%I #9 Sep 25 2024 09:59:47

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

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

%U 3,2,4,1,3,4,2,3,1,3,1,4,1,3,2,1,3,4,2

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

%C We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

%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 and run-compression (A376312):

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

%t First/@Split[Differences[Select[Range[100], !SquareFreeQ[#]&]]]

%Y For nonprime instead of squarefree numbers we have A037201, halved A373947.

%Y Before compressing we had A078147.

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

%Y For squarefree instead of nonsquarefree we have A376305, ones A376342.

%Y For prime-powers instead of nonsquarefree numbers we have A376308.

%Y A000040 lists the prime numbers, differences A001223.

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

%Y A003242 counts compressed compositions, ranks A333489.

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

%Y A013929 lists nonsquarefree numbers, differences A078147.

%Y A116861 counts partitions by compressed sum, by compressed length A116608.

%Y Cf. A007674, A053797, A053806, A072284, A112925, A120992, A274174, A373198, A375707, A376306, A376307, A376311.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 24 2024