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A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

%I A375738 #5 Sep 11 2024 10:07:12
%S A375738 2,3,6,7,11,12,13,14,15,18,19,20,21,22,23,24,29,30,31,34,35,38,39,40,
%T A375738 41,42,43,44,45,46,47,48,51,52,53,54,55,56,57,58,59,60,61,62,63,66,67,
%U A375738 68,69,70,71,72,73,74,75,76,77,78,79,80,83,84,85,86,87,88
%N A375738 Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.
%C A375738 Non-perfect-powers (A007916) are numbers with no proper integer roots.
%C A375738 An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.
%e A375738 The initial anti-runs are the following, whose minima are a(n):
%e A375738   (2)
%e A375738   (3,5)
%e A375738   (6)
%e A375738   (7,10)
%e A375738   (11)
%e A375738   (12)
%e A375738   (13)
%e A375738   (14)
%e A375738   (15,17)
%e A375738   (18)
%e A375738   (19)
%e A375738   (20)
%e A375738   (21)
%e A375738   (22)
%e A375738   (23)
%e A375738   (24,26,28)
%t A375738 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375738 Min/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most
%Y A375738 For composite numbers we have A005381, runs A008864 (except first term).
%Y A375738 For prime-powers we have A120430, runs A373673 (except first term).
%Y A375738 For squarefree numbers we have A373408, runs A072284.
%Y A375738 For nonsquarefree numbers we have A373410, runs A053806.
%Y A375738 For non-prime-powers we have A373575, runs A373676.
%Y A375738 For anti-runs of non-perfect-powers:
%Y A375738 - length: A375736
%Y A375738 - first: A375738 (this)
%Y A375738 - last: A375739
%Y A375738 - sum: A375737
%Y A375738 For runs of non-perfect-powers:
%Y A375738 - length: A375702
%Y A375738 - first: A375703
%Y A375738 - last: A375704
%Y A375738 - sum: A375705
%Y A375738 A001597 lists perfect-powers, differences A053289.
%Y A375738 A007916 lists non-perfect-powers, differences A375706.
%Y A375738 Cf. A007674, A045542, A046933, A061399, A216765, A251092, A373403, A373679, A375708, A375714.
%K A375738 nonn
%O A375738 1,1
%A A375738 _Gus Wiseman_, Sep 10 2024