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