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

1,2

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.

Links

Table of n, a(n) for n=1..56.

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

Example

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, ...

Mathematica

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

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.

Sequence in context: A004101 A003405 A153918 * A308909 A369356 A308958

Adjacent sequences: A376594 A376595 A376596 * A376598 A376599 A376600

Keyword

nonn,new

Author

Gus Wiseman, Oct 05 2024

Status

approved