A376651 Points of upward concavity in the sequence of composite numbers (A002808).

4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, 102, 112, 118, 123, 126, 131, 136, 146, 150, 162, 173, 176, 180, 185, 195, 200, 205, 210, 216, 219, 229, 242, 245, 249, 262, 267, 276, 280, 285, 292, 297, 302, 305, 310, 317, 320

Offset: 1

Author

Gus Wiseman, Oct 06 2024

Keywords

• nonn

Comments

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

Also positions of strict ascents in the first differences (A073783) of composite numbers (A002808).

Examples

			The composite numbers are (A002808):
  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 positive terms at (A376651):
  4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, ...
		

Links

• Dominic McCarty, Table of n, a(n) for n = 1..1000
• Gus Wiseman, Points of upward concavity in the sequence of composite numbers.

Crossrefs

The version for A000002 is A022297, negative A156242.

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

For first differences we had A065310 or A073783, ones A375929.

These are the positions of positive terms in A073445, negative A376652.

For prime instead of composite we have A258025, negative A258026.

For zero second differences (instead of positive) we have A376602.

For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376652 (concave-down).

Cf. A000961, A057820, A064113, A065890, A251092, A333254, A376604.

Programs

Mathematica

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],1]