A376601 Points of nonzero curvature in the sequence of non-prime-powers inclusive (A024619).

1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, 38, 40, 42, 46, 49, 53, 54, 60, 61, 63, 65, 67, 69, 71, 72, 78, 79, 81, 82, 83, 84, 85, 89, 91, 99, 101, 105, 106, 110, 111, 113, 115, 117, 118, 122, 124, 132, 134, 136, 138, 148

Offset: 1

Author

Gus Wiseman, Oct 05 2024

Keywords

• nonn

Comments

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

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, subtract 1 and shift left.

Examples

			The non-prime-powers inclusive (A024619) are:
  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 nonzero terms (A376601) at:
  1, 3, 4, 5, 6, 8, 12, 13, 16, 17, 19, 21, 23, 25, 27, 28, 32, 34, 35, 36, 37, ...
		

Links

• Gus Wiseman, Points of nonzero curvature in the non-prime-powers inclusive.

Crossrefs

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

These are the nonzeros of A376599.

The complement is A376600.

A000961 lists prime-powers inclusive, exclusive A246655.

A007916 lists non-perfect-powers.

A024619/A361102 list non-prime-powers inclusive.

A057820 gives first differences of 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), A376600 (inflections and undulations).

For nonzero curvature: A333214 (prime), A376603 (composite), A376588 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power).

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

Programs

Mathematica

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