A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 55, 56, 57, 58, 59, 62, 64, 66, 68, 70, 73, 74, 75, 76, 77, 80, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 107, 108, 109, 112, 114, 116

Offset: 1

Author

Gus Wiseman, Oct 05 2024

Keywords

• nonn

Comments

These are points at which the second differences (A376599) are zero.

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, add 1 to all terms.

Examples

			The non-prime-powers inclusive are (A024619):
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with zeros at (A376600):
  2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, ...
		

Links

• Gus Wiseman, Inflection and undulation points in the non-prime-powers inclusive.

Crossrefs

For first differences we had A375735, ones A375713(n)-1.

These are the zeros of A376599.

The complement is A376601.

A000961 lists prime-powers inclusive, exclusive A246655.

A001597 lists perfect-powers, complement A007916.

A024619/A361102 list non-prime-powers inclusive.

A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.

For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376601 (nonzero curvature).

For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).

Cf. A025475, A053707, A057820, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Programs

Mathematica

Join@@Position[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2],0]