A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163

Offset: 1

Author

Gus Wiseman, Oct 05 2024

Keywords

• nonn

Comments

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

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 nonzero terms at (A376603):
  2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...
		

Links

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

Crossrefs

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

These are the positions of nonzero terms in A073445.

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

For prime instead of composite we have A333214.

The complement is A376602.

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

For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).

For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).

Cf. A000961, A064113, A246655, A251092, A258025, A333254.

Programs

Mathematica

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