A376594 - OEIS

%I #5 Oct 05 2024 09:40:48

%S 5,11,12,13,17,19,20,25,33,37,39,40,41,47,53,57,62,70,71,76,81,82,83,

%T 88,92,93,96,98,103,109,113,118,123,130,131,133,137,139,146,149,154,

%U 155,156,161,165,168,169,174,179,180,183,187,188,189,193,201,211,213

%N Inflection and undulation points in the sequence of nonsquarefree numbers (A013929).

%C These are points at which the second differences (A376593) are zero.

%H Gus Wiseman, <a href="/A376594/a376594.png">Inflection and undulation points in the nonsquarefree numbers</a>.

%e The nonsquarefree numbers (A013929) are:

%e   4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, ...

%e with first differences (A078147):

%e   4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, 3, ...

%e with first differences (A376593):

%e   -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, 3, ...

%e with zeros (A376594) at:

%e   5, 11, 12, 13, 17, 19, 20, 25, 33, 37, 39, 40, 41, 47, 53, 57, 62, 70, 71, 76, ...

%t Join@@Position[Differences[Select[Range[100],!SquareFreeQ[#]&],2],0]

%Y The first differences were A078147.

%Y These are the zeros of A376593.

%Y The complement is A376595.

%Y A000040 lists the prime numbers, differences A001223.

%Y A005117 lists squarefree numbers, differences A076259.

%Y A013929 lists nonsquarefree numbers, differences A078147.

%Y A064113 lists positions of adjacent equal prime gaps.

%Y A114374 counts partitions into nonsquarefree numbers.

%Y For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376597 (prime-power), A376600 (non-prime-power).

%Y For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376593 (second differences), A376595 (nonzero curvature).

%Y Cf. A007674, A053797, A053806, A061398, A112926, A120992, A251092, A375707, A376312, A376590, A376593.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 04 2024