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A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

%I A373400 #8 Jun 10 2024 14:59:54
%S A373400 1,3,8,23,29,33,45,98,153,188,216,262,281,366,428,589,737,1182,1830,
%T A373400 1878,2190,2224,3076,3301,3384,3426,3643,3792,4521,4611,7969,8027,
%U A373400 8687,12541,14356,14861,15782,17005,19025,23282,30801,31544,33607,34201,34214,38589
%N A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).
%C A373400 The unsorted version is A073051.
%C A373400 A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.
%H A373400 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>
%e A373400 The maximal runs of composite numbers begin:
%e A373400    4
%e A373400    6
%e A373400    8   9  10
%e A373400   12
%e A373400   14  15  16
%e A373400   18
%e A373400   20  21  22
%e A373400   24  25  26  27  28
%e A373400   30
%e A373400   32  33  34  35  36
%e A373400   38  39  40
%e A373400   42
%e A373400   44  45  46
%e A373400   48  49  50  51  52
%e A373400   54  55  56  57  58
%e A373400   60
%e A373400   62  63  64  65  66
%e A373400   68  69  70
%e A373400   72
%e A373400   74  75  76  77  78
%e A373400   80  81  82
%e A373400   84  85  86  87  88
%e A373400   90  91  92  93  94  95  96
%e A373400   98  99 100
%e A373400 The a(n)-th rows are:
%e A373400    4
%e A373400    8   9  10
%e A373400   24  25  26  27  28
%e A373400   90  91  92  93  94  95  96
%e A373400  114 115 116 117 118 119 120 121 122 123 124 125 126
%e A373400  140 141 142 143 144 145 146 147 148
%e A373400  200 201 202 203 204 205 206 207 208 209 210
%t A373400 t=Length/@Split[Select[Range[10000],CompositeQ],#1+1==#2&]//Most;
%t A373400 Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
%Y A373400 The unsorted version is A073051, firsts of A176246.
%Y A373400 For squarefree runs we have the triple (1,3,5), firsts of A120992.
%Y A373400 For prime runs we have the triple (1,2,3), firsts of A175632.
%Y A373400 For squarefree antiruns we have A373128, firsts of A373127.
%Y A373400 For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
%Y A373400 For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
%Y A373400 For composite runs we have the triple (1,2,7), firsts of A373403.
%Y A373400 A000040 lists the primes, differences A001223.
%Y A373400 A002808 lists the composite numbers, differences A073783.
%Y A373400 A046933 counts composite numbers between primes.
%Y A373400 A065855 counts composite numbers up to n.
%Y A373400 Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.
%K A373400 nonn
%O A373400 1,2
%A A373400 _Gus Wiseman_, Jun 10 2024