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Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).
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%I #5 Oct 05 2024 14:00:22

%S 4,5,7,9,10,11,12,13,17,18,19,20,21,22,23,24,25,26,28,29,30,31,33,34,

%T 35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,

%U 59,60,62,63,64,65,66,68,69,70,71,73,74,75,76,77,78,79,80

%N Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

%C These are points at which the second differences (A376596) are nonzero.

%C Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

%H Gus Wiseman, <a href="/A376598/a376598.png">Points of nonzero curvature in the prime-powers inclusive</a>.

%e The prime-powers inclusive (A000961) are:

%e 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...

%e with first differences (A057820):

%e 1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...

%e with first differences (A376596):

%e 0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...

%e with nonzeros at (A376598):

%e 4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...

%t Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]

%Y The first differences were A057820, see also A376340.

%Y First differences are A376309.

%Y These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).

%Y The complement is A376597.

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

%Y A001597 lists perfect-powers, complement A007916.

%Y A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.

%Y `A064113 lists positions of adjacent equal prime gaps.

%Y For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).

%Y For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).

%Y Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 05 2024