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Run-sums of the sequence of first differences of squarefree numbers.
14

%I #10 Sep 21 2024 19:19:09

%S 2,2,2,3,1,2,2,6,2,6,2,2,2,2,2,2,2,3,1,4,6,2,2,1,3,2,2,2,2,1,3,2,3,1,

%T 2,2,6,2,2,4,2,2,2,2,2,2,2,3,1,3,1,4,2,2,2,1,3,2,2,2,2,1,3,2,3,1,2,2,

%U 6,2,6,1,3,2,2,2,2,2,3,1,2,2,6,2,2,1,3

%N Run-sums of the sequence of first differences of squarefree numbers.

%e The sequence of squarefree numbers (A005117) is:

%e 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...

%e The sequence of first differences (A076259) of squarefree numbers is:

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

%e with runs:

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

%e with sums A376307 (this sequence).

%t Total/@Split[Differences[Select[Range[100],SquareFreeQ]]]

%Y Run-sums of first differences of A005117.

%Y Before taking run-sums we had A076259, ones A375927.

%Y For the squarefree numbers themselves we have A373413.

%Y For prime instead of squarefree numbers we have A373822, halved A373823.

%Y For compression instead of run-sums we have A376305, ones A376342.

%Y For run-lengths instead of run-sums we have A376306.

%Y For prime-powers instead of squarefree numbers we have A376310.

%Y For positions of first appearances instead of run-sums we have A376311.

%Y A000040 lists the prime numbers, differences A001223.

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

%Y A003242 counts compressed or anti-run compositions, ranks A333489.

%Y A005117 lists squarefree numbers, differences A076259.

%Y A013929 lists nonsquarefree numbers, differences A078147.

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

%Y A274174 counts contiguous compositions, ranks A374249.

%Y Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A373197, A373198, A375707.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 21 2024