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Numbers k such that the k-th maximal antirun of squarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373127.
7

%I #10 Jun 11 2024 00:20:53

%S 1,3,8,10,19,162,1633,1853,2052,26661,46782,1080330,3138650

%N Numbers k such that the k-th maximal antirun of squarefree numbers has length different from all prior maximal antiruns. Sorted positions of first appearances in A373127.

%C The unsorted version is A373128.

%C An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

%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 squarefree numbers begin:

%e 1

%e 2

%e 3 5

%e 6

%e 7 10

%e 11 13

%e 14

%e 15 17 19 21

%e 22

%e 23 26 29

%e 30

%e 31 33

%e 34

%e 35 37

%e The a(n)-th rows are:

%e 1

%e 3 5

%e 15 17 19 21

%e 23 26 29

%e 47 51 53 55 57

%e 483 485 487 489 491 493

%t t=Length/@Split[Select[Range[10000],SquareFreeQ],#1+1!=#2&]//Most;

%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 The unsorted version is A373128, firsts of A373127.

%Y For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.

%Y For composite runs we have A373400, unsorted A073051.

%Y For prime antiruns we have A373402, unsorted A373401, firsts of A027833.

%Y For composite antiruns we have the triple (1,2,7), firsts of A373403.

%Y A005117 lists the squarefree numbers, first differences A076259.

%Y A013929 lists the nonsquarefree numbers, first differences A078147.

%Y Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Jun 10 2024