A375710 Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.

5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253

Offset: 1

Author

Gus Wiseman, Sep 09 2024

Keywords

• nonn

Comments

The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:

- A375709 (difference 1)

- A375710 (difference 2)

- A375711 (difference 3)

- A375712 (difference 4)

Examples

			The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.
		

Crossrefs

Positions of 2's in A078147.

For prime numbers we have A029707.

For nonprime numbers we appear to have A014689.

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.

A375707 counts squarefree numbers between consecutive nonsquarefree numbers.

Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.

Programs

Mathematica

Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]],2]

Formula

Complement of A375709 U A375711 U A375712.