A376594 Inflection and undulation points in the sequence of nonsquarefree numbers (A013929).

5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, 81, 82, 83, 88, 92, 93, 96, 98, 103, 109, 113, 118, 123, 130, 131, 133, 137, 139, 146, 149, 154, 155, 156, 161, 165, 168, 169, 174, 179, 180, 183, 187, 188, 189, 193, 201, 211, 213

Offset: 1

Author

Gus Wiseman, Oct 04 2024

Keywords

• nonn

Comments

These are points at which the second differences (A376593) are zero.

Examples

			The nonsquarefree numbers (A013929) are:
  4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, ...
with first differences (A078147):
  4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, ...
with first differences (A376593):
  -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, ...
with zeros (A376594) at:
  5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, ...
		

Links

• Gus Wiseman, Inflection and undulation points in the nonsquarefree numbers.

Crossrefs

The first differences were A078147.

These are the zeros of A376593.

The complement is A376595.

A000040 lists the prime numbers, differences A001223.

A005117 lists squarefree numbers, differences A076259.

A013929 lists nonsquarefree numbers, differences A078147.

A064113 lists positions of adjacent equal prime gaps.

A114374 counts partitions into nonsquarefree numbers.

For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376597 (prime-power), A376600 (non-prime-power).

For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376593 (second differences), A376595 (nonzero curvature).

Cf. A007674, A053797, A053806, A061398, A112926, A120992, A251092, A375707, A376312, A376590, A376593.

Programs

Mathematica

Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&],2],0]