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.
Original entry on oeis.org
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
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
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 composite antiruns we have the triple (1,2,7), firsts of
A373403.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf.
A006512,
A007674,
A049093,
A068781,
A072284,
A077641,
A174965,
A251092,
A373198,
A373408,
A373411.
-
t=Length/@Split[Select[Range[4,10000],PrimeQ],#1+2!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
A373573
Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.
Original entry on oeis.org
6, 1, 18, 8, 4, 2, 10, 52, 678
Offset: 1
The maximal antiruns of nonsquarefree numbers begin:
4 8
9 12 16 18 20 24
25 27
28 32 36 40 44
45 48
49
50 52 54 56 60 63
64 68 72 75
76 80
81 84 88 90 92 96 98
99
The a(n)-th rows are:
49
4 8
148 150 152
64 68 72 75
28 32 36 40 44
9 12 16 18 20 24
81 84 88 90 92 96 98
477 480 484 486 488 490 492 495
6345 6348 6350 6352 6354 6356 6358 6360 6363
For squarefree runs we have the triple (5,3,1), firsts of
A120992.
For prime runs we have the triple (1,3,2), firsts of
A175632.
For nonsquarefree runs we have
A373199 (assuming sorted), firsts of
A053797.
For composite antiruns we have the triple (2,7,1), firsts of
A373403.
Positions of first appearances in
A373409.
Cf.
A007674,
A025157,
A049094,
A061399,
A068781,
A072284,
A110969,
A251092,
A294242,
A373410,
A373412.
-
t=Length/@Split[Select[Range[10000],!SquareFreeQ[#]&],#1+1!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]
A373825
Position of first appearance of n in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.
Original entry on oeis.org
1, 2, 13, 11, 105, 57, 33, 69, 59, 29, 227, 129, 211, 341, 75, 321, 51, 45, 407, 313, 459, 301, 767, 1829, 413, 537, 447, 1113, 1301, 1411, 1405, 2865, 1709, 1429, 3471, 709, 2543, 5231, 1923, 679, 3301, 2791, 6555, 5181, 6345, 11475, 2491, 10633
Offset: 1
The runs of odd primes differing by 2 begin:
3 5 7
11 13
17 19
23
29 31
37
41 43
47
53
59 61
67
71 73
79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
3
2 2
1
2
1
2
1 1
2
1
2
1 1 1 1
2 2
1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, ...
with positions of first appearances a(n).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
-
t=Length/@Split[Length/@Split[Select[Range[3,10000], PrimeQ],#1+2==#2&]//Most]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#1]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]
A373200
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.
Original entry on oeis.org
1, 3, 8, 10, 19, 162, 1633, 1853, 2052, 26661, 46782, 1080330, 3138650
Offset: 1
The maximal antiruns of squarefree numbers begin:
1
2
3 5
6
7 10
11 13
14
15 17 19 21
22
23 26 29
30
31 33
34
35 37
The a(n)-th rows are:
1
3 5
15 17 19 21
23 26 29
47 51 53 55 57
483 485 487 489 491 493
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 composite antiruns we have the triple (1,2,7), firsts of
A373403.
Cf.
A006512,
A007674,
A049093,
A068781,
A072284,
A077641,
A174965,
A251092,
A373198,
A373408,
A373411.
-
t=Length/@Split[Select[Range[10000],SquareFreeQ],#1+1!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
A373824
Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.
Original entry on oeis.org
1, 2, 11, 13, 29, 33, 45, 51, 57, 59, 69, 75, 105, 129, 211, 227, 301, 313, 321, 341, 407, 413, 447, 459, 537, 679, 709, 767, 1113, 1301, 1405, 1411, 1429, 1439, 1709, 1829, 1923, 2491, 2543, 2791, 2865, 3301, 3471, 3641, 4199, 4611, 5181, 5231, 6345, 6555
Offset: 1
The runs of odd primes differing by 2 begin:
3 5 7
11 13
17 19
23
29 31
37
41 43
47
53
59 61
67
71 73
79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
3
2 2
1
2
1
2
1 1
2
1
2
1 1 1 1
2 2
1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3,...
with sorted positions of first appearances a(n).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
-
t=Length/@Split[Length/@Split[Select[Range[3,10000],PrimeQ],#1+2==#2&]];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
A373574
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.
Original entry on oeis.org
1, 2, 4, 6, 8, 10, 18, 52, 678
Offset: 1
The maximal antiruns of nonsquarefree numbers begin:
4 8
9 12 16 18 20 24
25 27
28 32 36 40 44
45 48
49
50 52 54 56 60 63
64 68 72 75
76 80
81 84 88 90 92 96 98
99
The a(n)-th rows are:
4 8
9 12 16 18 20 24
28 32 36 40 44
49
64 68 72 75
81 84 88 90 92 96 98
148 150 152
477 480 484 486 488 490 492 495
6345 6348 6350 6352 6354 6356 6358 6360 6363
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 composite antiruns we have the triple (1,2,7), firsts of
A373403.
Sorted positions of first appearances in
A373409.
Cf.
A007674,
A025157,
A049094,
A061399,
A068781,
A072284,
A077643,
A110969,
A251092,
A294242,
A373410,
A373412.
-
t=Length/@Split[Select[Range[100000],!SquareFreeQ[#]&],#1+1!=#2&];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
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