A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, 39, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 60, 61, 62, 65, 66, 67, 68, 69, 72, 74, 76, 78, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 99, 100, 101, 103, 105, 106, 107, 108

Offset: 1

Author

Gus Wiseman, Oct 05 2024

Keywords

• nonn

Comments

These are points at which the second differences (A073445) are zero.

Examples

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with zeros at (A376602):
  1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, ...
		

Links

• Wikipedia, Inflection point
• Gus Wiseman, Inflection and undulation points in the sequence of composite numbers.

Crossrefs

Partitions into composite numbers are counted by A023895, factorizations A050370.

For prime instead of composite we have A064113.

These are the positions of zeros in A073445.

For first differences we had A073783, ones A375929, complement A065890.

For concavity in primes we have A258025/A258026, weak A333230/A333231.

For upward concavity (instead of zero) we have A376651, downward A376652.

The complement is A376603.

For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376603 (nonzero curvature), A376651 (concave-up), A376652 (concave-down).

For inflection and undulation points: A064113 (prime), A376588 (non-perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).

Cf. A000040, A036263, A251092, A333214, A333216, A333254.

Programs

Mathematica

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