A175632 -id:A175632 - cp's OEIS frontend

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).

1, 3, 8, 23, 29, 33, 45, 98, 153, 188, 216, 262, 281, 366, 428, 589, 737, 1182, 1830, 1878, 2190, 2224, 3076, 3301, 3384, 3426, 3643, 3792, 4521, 4611, 7969, 8027, 8687, 12541, 14356, 14861, 15782, 17005, 19025, 23282, 30801, 31544, 33607, 34201, 34214, 38589

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A073051.

A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of composite numbers begin:
   4
   6
   8   9  10
  12
  14  15  16
  18
  20  21  22
  24  25  26  27  28
  30
  32  33  34  35  36
  38  39  40
  42
  44  45  46
  48  49  50  51  52
  54  55  56  57  58
  60
  62  63  64  65  66
  68  69  70
  72
  74  75  76  77  78
  80  81  82
  84  85  86  87  88
  90  91  92  93  94  95  96
  98  99 100
The a(n)-th rows are:
   4
   8   9  10
  24  25  26  27  28
  90  91  92  93  94  95  96
 114 115 116 117 118 119 120 121 122 123 124 125 126
 140 141 142 143 144 145 146 147 148
 200 201 202 203 204 205 206 207 208 209 210

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The unsorted version is A073051, firsts of A176246.
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 squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite runs 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.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

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

A373406 Sum of the n-th maximal run of odd primes differing by two.

Original entry on oeis.org

15, 24, 36, 23, 60, 37, 84, 47, 53, 120, 67, 144, 79, 83, 89, 97, 204, 216, 113, 127, 131, 276, 300, 157, 163, 167, 173, 360, 384, 396, 211, 223, 456, 233, 480, 251, 257, 263, 540, 277, 564, 293, 307, 624, 317, 331, 337, 696, 353, 359, 367, 373, 379, 383

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this run is given by A251092.
For this sequence we define a run to be an interval of positions at which consecutive terms differ by two. Normally, a run has consecutive terms differing by one, but odd prime numbers already differ by at least two.
Contains A054735 (sums of twin prime pairs) without its first two terms and A007510 (non-twin primes) as subsequences. - R. J. Mathar, Jun 07 2024

			Row-sums of:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
  83
  89
  97

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The partial sums are a subset of A071148 (partial sums of odd primes).
Functional neighbors: A025584, A054265, A067774, A251092 (or A175632), A373405, A373413, A373414.
A000040 lists the primes, differences A001223.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005117, A007053, A029707, A038664, A293697, A350842, A371201.

Total/@Split[Select[Range[3,100],PrimeQ],#1+2==#2&]//Most

A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

Original entry on oeis.org

1, 5, 7, 12, 18, 190, 28, 109, 40, 28195574, 53

Offset: 1

Gus Wiseman, Jun 14 2024

A run of a sequence (in this case A361102) is an interval of positions at which consecutive terms differ by one.
Are there only 9 terms?
From David A. Corneth, Jun 14 2024: (Start)
No. a(10) exists.
Between the prime 144115188075855859 and 144115188075855872 = 2^57 there are 12 non-prime-powers so a(12) exists. (End)

			The maximal runs of non-prime-powers begin:
   1
   6
  10
  12
  14  15
  18
  20  21  22
  24
  26
  28
  30
  33  34  35  36
  38  39  40
  42
  44  45  46
  48
  50  51  52
  54  55  56  57  58
  60

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For composite runs we have A073051, sorted A373400, firsts of A176246.
For squarefree runs we have firsts of A120992.
For prime-powers runs we have firsts of A174965.
For prime runs we have firsts of A251092 or A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199, firsts of A053797.
The sorted version is A373670.
For antiruns we have firsts of A373672.
For runs of non-prime-powers:
- length A110969
- min A373676
- max A373677
- sum A373678
A000961 lists the powers of primes (including 1).
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A007053, A008864, A014963, A027833, A038664, A054265, A067774, A356068, A373401, A373403, A373671.

q=Length/@Split[Select[Range[10000],!PrimePowerQ[#]&],#1+1==#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[y,Range[#1]]&];
Table[Position[q,k][[1,1]],{k,spna[q]}]

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

Gus Wiseman, Jun 10 2024

nonn

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.

			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

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

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.
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

Gus Wiseman, Jun 10 2024

nonn

The sorted version is A373574.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			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

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

For composite runs we have A073051, firsts of A176246, sorted A373400.
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 squarefree antiruns we have A373128, firsts of A373127, sorted A373200.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373401, firsts of A027833, sorted A373402.
For composite antiruns we have the triple (2,7,1), firsts of A373403.
Positions of first appearances in A373409.
The sorted version is A373574.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
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]}]

A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

Original entry on oeis.org

4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Labos Elemer

nonn

Numbers k such that prime(k+1) - prime(k) does not divide prime(k+1) + prime(k). These are the numbers k for which prime(k+1) - prime(k) > 2. - Thomas Ordowski, Mar 31 2022

If we prepend 1, the first differences are A251092 (see also A175632). The complement is A029707. - Gus Wiseman, Dec 03 2024

			prime(4) = 7, 6!+1 = 721 gives residue 1 when divided by prime(4)+2 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The first differences are A251092 except first term, run-lengths A373819.
The complement is A029707.
Runs of terms differing by one have lengths A027833, min A107770, max A155752.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A038664 finds the first prime gap of difference 2n.
A046933 counts composite numbers between primes.
A071148 gives partial sums of odd primes.
For prime runs: A001359, A005381, A006512, A025584, A067774, A373403.
Cf. A006560, A006562, A037201, A122535, A175632, A176246, A373820.

pnmQ[n_]:=Module[{p=Prime[n]},Mod[(p-1)!+1,p+2]==1]; Select[Range[ 100],pnmQ] (* Harvey P. Dale, Jun 24 2017 *)

isok(n) = (((prime(n)-1)! + 1) % (prime(n)+2)) == 1; \\ Michel Marcus, Dec 31 2013

Definition edited by Thomas Ordowski, Mar 31 2022

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

Gus Wiseman, Jun 10 2024

The unsorted version is A373128.

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

			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

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

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.
The unsorted version is A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For composite runs we have A373400, unsorted A073051.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
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[[#]]]&]

A179067 Orders of consecutive clusters of twin primes.

Original entry on oeis.org

1, 3, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Franz Vrabec, Jun 27 2010

For k>=1, 2k+4 consecutive primes P1, P2, ..., P2k+4 defining a cluster of twin primes of order k iff P2-P1 <> 2, P4-P3 = P6-P5 = ... = P2k+2 - P2k+1 = 2, P2k+4 - P2k+3 <> 2.

Also the lengths of maximal runs of terms differing by 2 in A029707 (leading index of twin primes), complement A049579. - Gus Wiseman, Dec 05 2024

			The twin prime cluster ((101,103),(107,109)) of order k=2 stems from the 2k+4 = 8 consecutive primes (89, 97, 101, 103, 107, 109, 113, 127) because 97-89 <> 2, 103-101 = 109-107 = 2, 127-113 <> 2.
From _Gus Wiseman_, Dec 05 2024: (Start)
The leading indices of twin primes are:
  2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, ...
with maximal runs of terms differing by 2:
  {2}, {3,5,7}, {10}, {13}, {17}, {20}, {26,28}, {33,35}, {41,43,45}, {49}, {52}, ...
with lengths a(n).
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A077800.
Cf. A001359, A111950, A087641.
Cf. A035789, A035790, A035791, A035792, A035793, A035794, A035795.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A006512 gives the greater of twin primes.
A029707 gives the leading index of twin primes, complement A049579.
A038664 finds the first prime gap of length 2n.
A046933 counts composite numbers between primes.
A000720, A006560, A006562, A014574, A037201, A107770, A122535, A155752, A175632, A251092.

R:= 1: count:= 1: m:= 0:
q:= 5: state:= 1:
while count < 100 do
 p:= nextprime(q);
 if state = 1 then
    if p-q = 2 then state:= 2; m:= m+1;
    else
      if m > 0 then R:= R,m; count:= count+1; fi;
      m:= 0
    fi
 else state:= 1;
 fi;
 q:= p
od:
R; # Robert Israel, Feb 07 2023

Length/@Split[Select[Range[2,100],Prime[#+1]-Prime[#]==2&],#2==#1+2&] (* Gus Wiseman, Dec 05 2024 *)

a(n)={my(o,P,L=vector(3));n++;forprime(p=o=3,,L=concat(L[2..3],-o+o=p);L[3]==2||next;L[1]==2&&(P=concat(P,p))&&next;n--||return(#P);P=[p])} \\ M. F. Hasler, May 04 2015

More terms from M. F. Hasler, May 04 2015

A256253 Number of successive odd nonprimes A014076 and number of successive odd primes A065091, interleaved.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 1, 6, 1, 1, 1, 2, 2, 4, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 2, 1, 2, 5, 1, 5, 1, 1, 2, 1, 1, 2, 2, 4, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 4, 1, 6, 1, 1, 2, 1, 1, 6, 1, 2, 1, 4, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2

Offset: 1

Omar E. Pol, Mar 30 2015

nonn

See also A256252 and A256262 which contain similar diagrams.

			Consider an irregular array in which the odd-indexed rows list successive odd nonprimes (A014076) and the even-indexed rows list successive odd primes (A065091), in the sequence of odd numbers (A005408), as shown below:
1;
3, 5, 7;
9;
11, 13;
15;
17; 19;
21,
23;
25, 27;
39, 31;
...
a(n) is the length of the n-th row.
.
Illustration of the first 16 regions of the diagram of the symmetric representation of odd nonprimes (A014076) and of odd primes (A065091):
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.           |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |   31
.           |_ _ _ _ _ _ _ _ _ _ _ _ _ _  | |   29
.           | | |_ _ _ _ _ _ _ _ _ _ _  | | |   23
.           | | | |_ _ _ _ _ _ _ _ _  | | | |   19
.           | | | |_ _ _ _ _ _ _ _  | | | | |   17
.           | | | | |_ _ _ _ _ _  | | | | | |   13
.           | | | | |_ _ _ _ _  | | | | | | |   11
.           | | | | | |_ _ _  | | | | | | | |    7
.           | | | | | |_ _  | | | | | | | | |    5
.   A014076 | | | | | |_  | | | | | | | | | |    3
.      1    | | | | | |_|_|_|_| | | | | | | | A065091
.      9    | | | | |_ _ _ _ _|_|_| | | | | |
.     15    | | | |_ _ _ _ _ _ _ _|_|_| | | |
.     21    | | |_ _ _ _ _ _ _ _ _ _ _|_| | |
.     25    | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
.     27    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|_|_|
.
a(n) is also the length of the n-th boundary segment in the zig-zag path of the above diagram, between the two types of numbers, as shown below for n = 1..10:
.
.                       |_ _ _
.                             |_ _
.                                 |_ _
.                                     |_
.                                       |
.                                       |_ _
.
The sequence begins:    1,3,1,2,1,2,1,1,2,2,...
.

Cf. A005408, A014076, A047846, A065091, A175632, A251092, A256252, A256134, A256262.

lista(nn) = {my(nb = 1, isp = 0); forstep (n=3, nn, 2, if (bitxor(isp, ! isprime(n)), nb++, print1(nb, ", "); nb = 1; isp = ! isp););} \\ Michel Marcus, May 25 2015

a(n) = A256252(n-1), n >= 3.

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

Gus Wiseman, Jun 10 2024

The unsorted version is A373573.
An antirun of a sequence (in this case A013929) is an interval of positions at which consecutive terms differ by more than one.
Is this sequence finite? Are there only 9 terms?

			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

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

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, firsts of A373127, unsorted A373128.
For composite runs we have A373400, firsts of A176246, unsorted A073051.
For prime antiruns we have A373402, firsts of A027833, unsorted A373401.
For composite antiruns we have the triple (1,2,7), firsts of A373403.
Sorted positions of first appearances in A373409.
The unsorted version is A373573.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
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[[#]]]&]

cp's OEIS Frontend

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).

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A373406 Sum of the n-th maximal run of odd primes differing by two.

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A373669 Least k such that the k-th maximal run of non-prime-powers has length n. Position of first appearance of n in A110969, and the sequence ends if there is none.

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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.

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A373573 Least k such that the k-th maximal antirun of nonsquarefree numbers has length n. Position of first appearance of n in A373409.

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A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

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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.

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A179067 Orders of consecutive clusters of twin primes.

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