A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Author

Gus Wiseman, Oct 05 2024

Keywords

• nonn

Comments

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

Examples

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with nonzeros at (A376598):
  4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...
		

Links

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

Crossrefs

The first differences were A057820, see also A376340.

First differences are A376309.

These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).

The complement is A376597.

A000961 lists prime-powers inclusive, exclusive A246655.

A001597 lists perfect-powers, complement A007916.

A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.

`A064113 lists positions of adjacent equal prime gaps.

For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).

For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).

Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Programs

Mathematica

Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]