A376589 - OEIS

A376589

Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, 106, 107, 109, 110, 111, 112, 126, 127, 150, 151, 176, 177, 195, 196, 203, 204, 220, 221, 232, 233, 264, 265, 298, 299, 316, 317, 333, 334, 371, 372, 411, 412, 453, 454, 480, 481, 496

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OFFSET

1,2

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..57.

Gus Wiseman, Points of nonzero curvature 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 nonzeros at (A376589):

1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, ...

MATHEMATICA

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

Join@@Position[Sign[Differences[Select[Range[1000], radQ], 2]], 1|-1]

CROSSREFS

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

These are the positions of nonzeros in A376562, complement A376588.

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.

For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376588 (inflection and undulation points).

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

Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A182853, A294068, A333254, A336416, A376599.

Sequence in context: A167795 A322068 A138048 * A057762 A109511 A018339

Adjacent sequences: A376586 A376587 A376588 * A376590 A376591 A376592

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 03 2024

STATUS

approved