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A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

%I A376589 #9 Oct 04 2024 11:18:04
%S A376589 1,2,4,5,10,11,18,20,23,24,26,27,38,39,52,53,68,69,86,87,106,107,109,
%T A376589 110,111,112,126,127,150,151,176,177,195,196,203,204,220,221,232,233,
%U A376589 264,265,298,299,316,317,333,334,371,372,411,412,453,454,480,481,496
%N A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).
%C A376589 These are points at which the second differences (A376562) are nonzero.
%C A376589 Non-perfect-powers (A007916) are numbers without a proper integer root.
%H A376589 Gus Wiseman, <a href="/A376589/a376589.png">Points of nonzero curvature in the non-perfect-powers</a>.
%e A376589 The non-perfect powers (A007916) are:
%e A376589   2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
%e A376589 with first differences (A375706):
%e A376589   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, ...
%e A376589 with first differences (A376562):
%e A376589   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, ...
%e A376589 with nonzeros at (A376589):
%e A376589   1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, ...
%t A376589 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A376589 Join@@Position[Sign[Differences[Select[Range[1000],radQ],2]],1|-1]
%Y A376589 For first differences we had A375706, ones A375740, complement A375714.
%Y A376589 These are the positions of nonzeros in A376562, complement A376588.
%Y A376589 Runs of non-perfect-powers:
%Y A376589 - length: A375702 = A053289(n+1) - 1
%Y A376589 - first: A375703 (same as A216765 with 2 exceptions)
%Y A376589 - last: A375704 (same as A045542 with 8 removed)
%Y A376589 - sum: A375705
%Y A376589 A000961 lists prime-powers inclusive, exclusive A246655.
%Y A376589 A007916 lists non-perfect-powers, complement A001597.
%Y A376589 A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
%Y A376589 For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376588 (inflection and undulation points).
%Y A376589 For second differences: A064113 (prime), A376602 (composite), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
%Y A376589 Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A182853, A294068, A333254, A336416, A376599.
%K A376589 nonn
%O A376589 1,2
%A A376589 _Gus Wiseman_, Oct 03 2024