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A373402
Numbers k such that the k-th maximal antirun of prime numbers > 3 has length different from all prior maximal antiruns. Sorted list of positions of first appearances in A027833.
10
1, 2, 4, 6, 8, 10, 21, 24, 30, 35, 40, 41, 46, 50, 69, 82, 131, 140, 185, 192, 199, 210, 248, 251, 271, 277, 325, 406, 423, 458, 645, 748, 811, 815, 826, 831, 987, 1053, 1109, 1426, 1456, 1590, 1629, 1870, 1967, 2060, 2371, 2607, 2920, 2946, 3564, 3681, 4119
OFFSET
1,2
COMMENTS
The unsorted version is A373401.
For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.
EXAMPLE
The maximal antiruns of prime numbers > 3 begin:
5
7 11
13 17
19 23 29
31 37 41
43 47 53 59
61 67 71
73 79 83 89 97 101
103 107
109 113 127 131 137
139 149
151 157 163 167 173 179
The a(n)-th rows begin:
5
7 11
19 23 29
43 47 53 59
73 79 83 89 97 101
109 113 127 131 137
MATHEMATICA
t=Length/@Split[Select[Range[4, 10000], PrimeQ], #1+2!=#2&]//Most;
Select[Range[Length[t]], FreeQ[Take[t, #-1], t[[#]]]&]
CROSSREFS
For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For squarefree antiruns: A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051.
The unsorted version is A373401, firsts of A027833.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Sequence in context: A157006 A221991 A279254 * A212495 A083490 A229363
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 10 2024
STATUS
approved