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A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.

%I A373409 #9 Jun 12 2024 17:21:28
%S A373409 2,6,2,5,2,1,6,4,2,7,1,5,2,2,1,4,4,3,6,2,2,4,7,5,7,1,1,6,6,2,3,4,7,3,
%T A373409 3,5,1,3,1,3,2,2,3,5,5,7,1,5,7,5,1,8,4,2,5,2,2,3,3,1,7,3,4,7,1,5,2,5,
%U A373409 2,6,7,6,7,5,1,2,3,5,6,4,1,3,5,7,2,3,2
%N A373409 Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one.
%C A373409 An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
%C A373409 Conjecture: The maximum is 9, and there is no antirun of more than 9 nonsquarefree numbers. Confirmed up to 100,000,000.
%H A373409 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>
%e A373409 Row-lengths of:
%e A373409    4   8
%e A373409    9  12  16  18  20  24
%e A373409   25  27
%e A373409   28  32  36  40  44
%e A373409   45  48
%e A373409   49
%e A373409   50  52  54  56  60  63
%e A373409   64  68  72  75
%e A373409   76  80
%e A373409   81  84  88  90  92  96  98
%e A373409   99
%e A373409 The first maximal antirun of length 9 is the following, shown with prime indices:
%e A373409   6345: {2,2,2,3,15}
%e A373409   6348: {1,1,2,9,9}
%e A373409   6350: {1,3,3,31}
%e A373409   6352: {1,1,1,1,78}
%e A373409   6354: {1,2,2,71}
%e A373409   6356: {1,1,4,49}
%e A373409   6358: {1,5,7,7}
%e A373409   6360: {1,1,1,2,3,16}
%e A373409   6363: {2,2,4,26}
%t A373409 Length/@Split[Select[Range[1000],!SquareFreeQ[#]&],#1+1!=#2&]//Most
%Y A373409 Positions of first appearances are A373573, sorted A373574.
%Y A373409 Functional neighbors: A027833, A053797, A068781, A373127, A373403, A373410, A373412.
%Y A373409 A005117 lists the squarefree numbers, first differences A076259.
%Y A373409 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A373409 Cf. A025157, A049093, A061398, A061399, A077641, A077643, A110969, A174965, A294242, A350842, A373198.
%K A373409 nonn
%O A373409 1,1
%A A373409 _Gus Wiseman_, Jun 06 2024