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

1,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Wikipedia, Inflection point

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

Example

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, ...

Maple

Cons:= remove(isprime, [seq(i, i=4..1000)]):
D1:= Cons[2..-1]-Cons[1..-2]:
D2:= D1[2..-1]-D1[1..-2]:
select(i -> D2[i]=0, [$1..nops(D2)]); # Robert Israel, Jun 07 2026

Mathematica

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

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).

Contains A336407.

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

Sequence in context: A024323 A118820 A117521 * A190053 A327209 A285590

Adjacent sequences: A376599 A376600 A376601 * A376603 A376604 A376605

Keyword

nonn,changed

Author

Gus Wiseman, Oct 05 2024

Status

approved