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A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).

%I A376588 #9 Oct 03 2024 11:59:56
%S A376588 3,6,7,8,9,12,13,14,15,16,17,19,21,22,25,28,29,30,31,32,33,34,35,36,
%T A376588 37,40,41,42,43,44,45,46,47,48,49,50,51,54,55,56,57,58,59,60,61,62,63,
%U A376588 64,65,66,67,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84
%N A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).
%C A376588 These are points at which the second differences (A376562) are zero.
%C A376588 Non-perfect-powers (A007916) are numbers without a proper integer root.
%H A376588 Gus Wiseman, <a href="/A376588/a376588.png">Inflection and undulation points in the non-perfect-powers</a>.
%e A376588 The non-perfect powers (A007916) are:
%e A376588   2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
%e A376588 with first differences (A375706):
%e A376588   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 A376588 with first differences (A376562):
%e A376588   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 A376588 with zeros at (A376588):
%e A376588   3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, ...
%t A376588 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A376588 Join@@Position[Differences[Select[Range[100],radQ],2],0]
%Y A376588 The version for A000002 is empty.
%Y A376588 For first differences we had A375706, ones A375740, complement A375714.
%Y A376588 Positions of zeros in A376562, complement A376589.
%Y A376588 Runs of non-perfect-powers:
%Y A376588 - length: A375702 = A053289(n+1) - 1
%Y A376588 - first: A375703 (same as A216765 with 2 exceptions)
%Y A376588 - last: A375704 (same as A045542 with 8 removed)
%Y A376588 - sum: A375705
%Y A376588 A000961 lists prime-powers inclusive, exclusive A246655.
%Y A376588 A007916 lists non-perfect-powers, complement A001597.
%Y A376588 A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
%Y A376588 A333254 gives run-lengths of differences between consecutive primes.
%Y A376588 For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376589 (nonzero curvature).
%Y A376588 For second differences: A064113 (prime), A376602 (composite), {} (perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power inclusive), A376600 (non-prime-power inclusive).
%Y A376588 Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A294068, A336416, A361102, A376599, A376590.
%K A376588 nonn
%O A376588 1,1
%A A376588 _Gus Wiseman_, Oct 03 2024