A376342 Positions of 1's in the run-compression (A376305) of the first differences (A076259) of the squarefree numbers (A005117).

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 51, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 124, 126, 128, 130

Offset: 1

Author

Gus Wiseman, Sep 24 2024

Keywords

• nonn

Comments

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).

Examples

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with run-compression (A376305):
  1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, ...
with ones at (A376342):
  1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, ...
		

Crossrefs

Before compressing we had A076259.

Positions of 1's in A376305.

The version for nonsquarefree numbers gives positions of ones in A376312.

For prime instead of squarefree numbers we have A376343.

A000040 lists the prime numbers, differences A001223.

A000961 and A246655 list prime-powers, differences A057820.

A003242 counts compressed compositions, ranks A333489.

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

A013929 lists nonsquarefree numbers, differences A078147.

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

A274174 counts contiguous compositions, ranks A374249.

Cf. A007674, A037201, A053797, A053806, A061398, A072284, A120992, A373198, A376306, A376307, A376308, A376311.

Programs

Mathematica

Join@@Position[First /@ Split[Differences[Select[Range[100],SquareFreeQ]]],1]