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Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).
8

%I #6 Oct 05 2024 14:00:18

%S 2,7,9,10,11,14,15,18,20,22,24,26,29,30,31,33,39,41,43,44,45,47,48,50,

%T 51,52,55,56,57,58,59,62,64,66,68,70,73,74,75,76,77,80,86,87,88,90,92,

%U 93,94,95,96,97,98,100,102,103,104,107,108,109,112,114,116

%N Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

%C These are points at which the second differences (A376599) are zero.

%C Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, add 1 to all terms.

%H Gus Wiseman, <a href="/A376600/a376600.png">Inflection and undulation points in the non-prime-powers inclusive</a>.

%e The non-prime-powers inclusive are (A024619):

%e 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...

%e with first differences (A375735):

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

%e with first differences (A376599):

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

%e with zeros at (A376600):

%e 2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, ...

%t Join@@Position[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2],0]

%Y For first differences we had A375735, ones A375713(n)-1.

%Y These are the zeros of A376599.

%Y The complement is A376601.

%Y A000961 lists prime-powers inclusive, exclusive A246655.

%Y A001597 lists perfect-powers, complement A007916.

%Y A024619/A361102 list non-prime-powers inclusive.

%Y A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.

%Y For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376601 (nonzero curvature).

%Y For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).

%Y Cf. A025475, A053707, A057820, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 05 2024