%I #6 Dec 02 2024 10:10:34

%S 1,0,1,2,0,0,2,0,1,0,1,3,2,1,0,1,0,0,1,0,1,2,1,0,2,2,1,0,2,0,1,3,0,1,

%T 3,0,0,0,1,2,2,2,0,2,0,2,0,0,0,2,2,0,1,3,2,0,0,0,0,2,2,1,0,2,0,1,0,1,

%U 0,2,2,3,0,1,2,0,0,3,2,0,2,3,3,2,0,1,2

%N Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

%C All terms are 0, 1, 2, or 3 (cf. A078147).

%C The inclusive version is a(n) + 2.

%C The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

%e The composite numbers counted by a(n) form the following set partition of A120944:

%e {6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

%t v=Select[Range[100],!SquareFreeQ[#]&];

%t Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

%Y For prime (instead of nonsquarefree) we have A046933.

%Y For squarefree (instead of nonsquarefree) we have A076259(n)-1.

%Y For prime power (instead of nonsquarefree) we have A093555.

%Y For prime instead of composite we have A236575.

%Y For nonprime prime power (instead of nonsquarefree) we have A378456.

%Y For perfect power (instead of nonsquarefree) we have A378614, primes A080769.

%Y A002808 lists the composite numbers.

%Y A005117 lists the squarefree numbers, differences A076259.

%Y A013929 lists the nonsquarefree numbers, differences A078147.

%Y A073247 lists squarefree numbers with nonsquarefree neighbors.

%Y A120944 lists squarefree composite numbers.

%Y A377432 counts perfect-powers between primes, zeros A377436.

%Y A378369 gives distance to the next nonsquarefree number (A120327).

%Y Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

%K nonn

%O 1,4

%A _Gus Wiseman_, Dec 02 2024