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A375739 Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

%I A375739 #5 Sep 11 2024 10:07:01
%S A375739 2,5,6,10,11,12,13,14,17,18,19,20,21,22,23,28,29,30,33,34,37,38,39,40,
%T A375739 41,42,43,44,45,46,47,50,51,52,53,54,55,56,57,58,59,60,61,62,65,66,67,
%U A375739 68,69,70,71,72,73,74,75,76,77,78,79,82,83,84,85,86,87,88
%N A375739 Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.
%C A375739 Non-perfect-powers (A007916) are numbers with no proper integer roots.
%C A375739 An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.
%C A375739 Also non-perfect-powers x such that x + 1 is also a non-perfect-power.
%e A375739 The initial anti-runs are the following, whose maxima are a(n):
%e A375739   (2)
%e A375739   (3,5)
%e A375739   (6)
%e A375739   (7,10)
%e A375739   (11)
%e A375739   (12)
%e A375739   (13)
%e A375739   (14)
%e A375739   (15,17)
%e A375739   (18)
%e A375739   (19)
%e A375739   (20)
%e A375739   (21)
%e A375739   (22)
%e A375739   (23)
%e A375739   (24,26,28)
%t A375739 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375739 Max/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most
%t A375739 - or -
%t A375739 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375739 Select[Range[100],radQ[#]&&radQ[#+1]&]
%Y A375739 For nonprime numbers we have A068780, runs A006093 with 2 removed.
%Y A375739 For squarefree numbers we have A007674, runs A373415.
%Y A375739 For nonsquarefree numbers we have A068781, runs A072284 minus 1 and shifted.
%Y A375739 For prime-powers we have A006549, runs A373674.
%Y A375739 For non-prime-powers we have A255346, runs A373677.
%Y A375739 For anti-runs of non-perfect-powers:
%Y A375739 - length: A375736
%Y A375739 - first: A375738
%Y A375739 - last: A375739 (this)
%Y A375739 - sum: A375737
%Y A375739 For runs of non-perfect-powers:
%Y A375739 - length: A375702
%Y A375739 - first: A375703
%Y A375739 - last: A375704
%Y A375739 - sum: A375705
%Y A375739 A001597 lists perfect-powers, differences A053289.
%Y A375739 A007916 lists non-perfect-powers, differences A375706.
%Y A375739 Cf. A045542, A046933, A053797, A216765, A373576, A373679, A375714.
%K A375739 nonn
%O A375739 1,1
%A A375739 _Gus Wiseman_, Sep 10 2024