A376595 Points of nonzero curvature in the sequence of nonsquarefree numbers (A013929).

1, 2, 3, 4, 6, 7, 8, 9, 10, 14, 15, 16, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 38, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 77, 78, 79, 80, 84, 85, 86, 87, 89, 90, 91

Offset: 1

Author

Gus Wiseman, Oct 04 2024

Keywords

• nonn

Comments

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

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 nonzeros (A376594) at:
  1, 2, 3, 4, 6, 7, 8, 9, 10, 14, 15, 16, 18, 21, 22, 23, 24, 26, 27, 28, 29, 30, ...
		

Links

• Gus Wiseman, Points of nonzero curvature in the nonsquarefree numbers.

Crossrefs

The first differences were A078147.

These are the nonzeros of A376593.

The complement is A376594.

A000040 lists the prime numbers, differences A001223.

A005117 lists squarefree numbers, differences A076259.

A013929 lists nonsquarefree numbers, differences A078147.

A114374 counts integer partitions into nonsquarefree numbers.

For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376598 (prime-power), A376601 (non-prime-power).

For nonsquarefree numbers: A078147 (first differences), A376593 (second differences), A376594 (inflections and undulations).

Cf. A007674, A036263, A053797, A053806, A061398, A120992, A373198, A375707, A376306, A376312, A376590.

Programs

Mathematica

Join@@Position[Sign[Differences[Select[Range[100],!SquareFreeQ[#]&],2]],1|-1]