A376588 - OEIS

A376588

Inflection and undulation points in the sequence of non-perfect-powers (A007916).

3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

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OFFSET

1,1

COMMENTS

These are points at which the second differences (A376562) are zero.

Non-perfect-powers (A007916) are numbers without a proper integer root.

LINKS

Table of n, a(n) for n=1..66.

Gus Wiseman, Inflection and undulation points in the non-perfect-powers.

EXAMPLE

The non-perfect powers (A007916) are:

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...

with first differences (A375706):

1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...

with first differences (A376562):

1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...

with zeros at (A376588):

3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, ...

MATHEMATICA

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

Join@@Position[Differences[Select[Range[100], radQ], 2], 0]

CROSSREFS

The version for A000002 is empty.

For first differences we had A375706, ones A375740, complement A375714.

Positions of zeros in A376562, complement A376589.

Runs of non-perfect-powers:

- length: A375702 = A053289(n+1) - 1

- first: A375703 (same as A216765 with 2 exceptions)

- last: A375704 (same as A045542 with 8 removed)

- sum: A375705

A000961 lists prime-powers inclusive, exclusive A246655.

A007916 lists non-perfect-powers, complement A001597.

A305631 counts integer partitions into non-perfect-powers, factorizations A322452.

A333254 gives run-lengths of differences between consecutive primes.

For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376589 (nonzero curvature).

For second differences: A064113 (prime), A376602 (composite), {} (perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power inclusive), A376600 (non-prime-power inclusive).

Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A294068, A336416, A361102, A376599, A376590.

Sequence in context: A047559 A288742 A113826 * A053030 A189822 A051205

Adjacent sequences: A376585 A376586 A376587 * A376589 A376590 A376591

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 03 2024

STATUS

approved