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Run-sums of first differences (A078147) of nonsquarefree numbers (A013929).
5

%I #10 Sep 27 2024 12:00:13

%S 4,1,3,4,4,4,1,2,1,16,1,3,2,6,4,3,1,8,3,1,4,1,3,4,4,4,2,2,16,1,3,1,3,

%T 2,2,4,3,1,8,3,1,4,1,3,4,4,4,1,2,1,3,1,12,1,3,4,4,4,3,1,16,1,3,4,4,4,

%U 2,3,3,4,8,1,3,4,4,3,1,3,1,8,1,3,4,1,3,4

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

%C Does the image include all positive integers? I have only verified this up to 8.

%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 sums (A376264):

%e 4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, ...

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

%Y Before taking run-sums we had A078147.

%Y For nonprime instead of nonsquarefree numbers we have A373822.

%Y Positions of first appearances are A376265, sorted A376266.

%Y For run-lengths instead of run-sums we have A376267.

%Y For squarefree instead of nonsquarefree we have A376307.

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

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

%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 Cf. A053797, A053806, A061398, A072284, A120992, A373197, A373413, A375707, A376305, A376306, A376311.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 26 2024