A376652 Points of downward concavity in the sequence of composite numbers (A002808).

2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, 104, 114, 119, 124, 127, 132, 138, 148, 152, 163, 174, 178, 181, 187, 196, 201, 206, 212, 217, 221, 230, 243, 247, 250, 263, 268, 278, 281, 286, 293, 298, 303, 306, 311, 318, 321

Offset: 1

Author

Gus Wiseman, Oct 06 2024

Keywords

• nonn

Comments

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

Also positions of strict descents 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 second 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 negative terms at (A376651):
  2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Gus Wiseman, Points of downward concavity in the sequence of composite numbers.

Crossrefs

The version for A000002 is A156242, positive A022297.

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 negative terms in A073445, positive A376651.

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

For zero second differences instead of negative we have A376602.

For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376651 (concave-up).

Cf. A000961, A007916, A057820, A064113, A065890, A251092, A333254, A376604.

Programs

Maple

Comps:= remove(isprime, [seq(i,i=4..1000)]):
D1:= Comps[2..-1]-Comps[1..-2]:
D2:= D1[2..-1]-D1[1..-2]:
select(t -> D2[t] < 0, [$1..nops(D2)]); # Robert Israel, Nov 06 2024

Mathematica

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