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Inflection and undulation points in the sequence of squarefree numbers (A005117).
17

%I #6 Oct 05 2024 09:40:57

%S 1,4,9,11,12,14,16,18,21,24,27,32,33,35,40,43,48,53,55,56,58,62,65,68,

%T 71,79,84,87,96,98,99,101,103,107,110,113,118,120,121,123,128,131,134,

%U 137,142,144,145,147,152,153,155,158,163,165,166,172,175,179,184

%N Inflection and undulation points in the sequence of squarefree numbers (A005117).

%C These are points at which the second differences (A376590) are zero.

%H Gus Wiseman, <a href="/A376591/a376591.png">Inflection and undulation points in the squarefree numbers</a>.

%e The squarefree numbers (A005117) are:

%e 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...

%e with first differences (A076259):

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

%e with first differences (A376590):

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

%e with zeros at (A376591):

%e 1, 4, 9, 11, 12, 14, 16, 18, 21, 24, 27, 32, 33, 35, 40, 43, 48, 53, 55, 56, 58, ...

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

%Y The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.

%Y These are the zeros of A376590.

%Y The complement is A376592.

%Y A000040 lists the prime numbers, differences A001223.

%Y A005117 lists squarefree numbers, complement A013929 (differences A078147).

%Y A073576 counts integer partitions into squarefree numbers, factorizations A050320.

%Y For inflections and undulations: A064113 (prime), A376602 (composite), A376588 (non-perfect-power), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).

%Y For squarefree numbers: A076259 (first differences), A376590 (second differences), A376592 (nonzero curvature).

%Y Cf. A007674, A036263, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A333254, A376593, A376655.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 04 2024