A376592 Points of nonzero curvature in the sequence of squarefree numbers (A005117).

2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 85, 86, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Author

Gus Wiseman, Oct 04 2024

Keywords

• nonn

Comments

These are points at which the second differences (A376590) are nonzero.

Examples

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
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, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...
with nonzeros at (A376591):
  2, 3, 5, 6, 7, 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, 28, 29, 30, 31, 34, 36, ...
		

Links

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

Crossrefs

The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.

These are the nonzeros of A376590.

The complement is A376591.

A000040 lists the prime numbers, differences A001223.

A005117 lists squarefree numbers, complement A013929 (differences A078147).

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).

For squarefree numbers: A076259 (first differences), A376590 (second differences), A376591 (inflection and undulation points).

Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A373198, A376655.

Programs

Mathematica

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