A376591 Inflection and undulation points in the sequence of squarefree numbers (A005117).

1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, 62, 65, 68, 71, 79, 84, 87, 96, 98, 99, 101, 103, 107, 110, 113, 118, 120, 121, 123, 128, 131, 134, 137, 142, 144, 145, 147, 152, 153, 155, 158, 163, 165, 166, 172, 175, 179, 184

Offset: 1

Author

Gus Wiseman, Oct 04 2024

Keywords

• nonn

Comments

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

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 zeros at (A376591):
 1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, ...
		

Links

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

Crossrefs

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

These are the zeros of A376590.

The complement is A376592.

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 inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).

For squarefree numbers: A076259 (first differences), A376590 (second differences), A376592 (nonzero curvature).

Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A376593, A376655.

Programs

Mathematica

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