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A373574 Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.

%I A373574 #5 Jun 11 2024 09:36:39
%S A373574 1,2,4,6,8,10,18,52,678
%N A373574 Numbers k such that the k-th maximal antirun of nonsquarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373409.
%C A373574 The unsorted version is A373573.
%C A373574 An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
%C A373574 Is this sequence finite? Are there only 9 terms?
%H A373574 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A373574 The maximal antiruns of nonsquarefree numbers begin:
%e A373574    4   8
%e A373574    9  12  16  18  20  24
%e A373574   25  27
%e A373574   28  32  36  40  44
%e A373574   45  48
%e A373574   49
%e A373574   50  52  54  56  60  63
%e A373574   64  68  72  75
%e A373574   76  80
%e A373574   81  84  88  90  92  96  98
%e A373574   99
%e A373574 The a(n)-th rows are:
%e A373574      4    8
%e A373574      9   12   16   18   20   24
%e A373574     28   32   36   40   44
%e A373574     49
%e A373574     64   68   72   75
%e A373574     81   84   88   90   92   96   98
%e A373574    148  150  152
%e A373574    477  480  484  486  488  490  492  495
%e A373574   6345 6348 6350 6352 6354 6356 6358 6360 6363
%t A373574 t=Length/@Split[Select[Range[100000],!SquareFreeQ[#]&],#1+1!=#2&];
%t A373574 Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
%Y A373574 For squarefree runs we have the triple (1,3,5), firsts of A120992.
%Y A373574 For prime runs we have the triple (1,2,3), firsts of A175632.
%Y A373574 For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
%Y A373574 For squarefree antiruns: A373200, firsts of A373127, unsorted A373128.
%Y A373574 For composite runs we have A373400, firsts of A176246, unsorted A073051.
%Y A373574 For prime antiruns we have A373402, firsts of A027833, unsorted A373401.
%Y A373574 For composite antiruns we have the triple (1,2,7), firsts of A373403.
%Y A373574 Sorted positions of first appearances in A373409.
%Y A373574 The unsorted version is A373573.
%Y A373574 A005117 lists the squarefree numbers, first differences A076259.
%Y A373574 A013929 lists the nonsquarefree numbers, first differences A078147.
%Y A373574 Cf. A007674, A025157, A049094, A061399, A068781, A072284, A077643, A110969, A251092, A294242, A373410, A373412.
%K A373574 nonn,more
%O A373574 1,2
%A A373574 _Gus Wiseman_, Jun 10 2024