A373400 - OEIS

%I #8 Jun 10 2024 14:59:54

%S 1,3,8,23,29,33,45,98,153,188,216,262,281,366,428,589,737,1182,1830,

%T 1878,2190,2224,3076,3301,3384,3426,3643,3792,4521,4611,7969,8027,

%U 8687,12541,14356,14861,15782,17005,19025,23282,30801,31544,33607,34201,34214,38589

%N 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 The unsorted version is A073051.

%C A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by 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 runs of composite numbers begin:

%e    4

%e    6

%e    8   9  10

%e   12

%e   14  15  16

%e   18

%e   20  21  22

%e   24  25  26  27  28

%e   30

%e   32  33  34  35  36

%e   38  39  40

%e   42

%e   44  45  46

%e   48  49  50  51  52

%e   54  55  56  57  58

%e   60

%e   62  63  64  65  66

%e   68  69  70

%e   72

%e   74  75  76  77  78

%e   80  81  82

%e   84  85  86  87  88

%e   90  91  92  93  94  95  96

%e   98  99 100

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

%e    4

%e    8   9  10

%e   24  25  26  27  28

%e   90  91  92  93  94  95  96

%e  114 115 116 117 118 119 120 121 122 123 124 125 126

%e  140 141 142 143 144 145 146 147 148

%e  200 201 202 203 204 205 206 207 208 209 210

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

%t Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

%Y The unsorted version is A073051, firsts of A176246.

%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 squarefree antiruns we have A373128, firsts of A373127.

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

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

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

%Y A000040 lists the primes, differences A001223.

%Y A002808 lists the composite numbers, differences A073783.

%Y A046933 counts composite numbers between primes.

%Y A065855 counts composite numbers up to n.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 10 2024