A376597 Inflection and undulation points in the sequence of prime-powers inclusive (A000961).

1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, 136, 141, 185, 190, 211, 214, 221, 226, 268, 292, 301, 302, 322, 374, 394, 423, 456, 463, 502, 503, 547, 559, 560, 593, 604, 640, 646, 663, 671, 675, 710, 726, 727, 746, 754, 755

Offset: 1

Author

Gus Wiseman, Oct 05 2024

Keywords

• nonn

Comments

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

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

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 zeros (A376597) at:
  1, 2, 3, 6, 8, 14, 15, 16, 27, 32, 50, 61, 67, 72, 85, 92, 93, 124, 129, 132, ...
		

Links

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

Crossrefs

The first differences were A057820, see also A053707, A376340.

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

The complement is A376598.

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), A376596 (second differences), A376598 (nonzero curvature).

For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).

Cf. A025475, A053707, A069623, A093555, A174965, A361102, A376653, A376654.

Programs

Mathematica

Join@@Position[Differences[Select[Range[1000],#==1||PrimePowerQ[#]&],2],0]