A373200 -id:A373200 - cp's OEIS frontend

A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4, 10, 2, 3, 6, 4, 2, 2, 7, 2, 6, 3

Offset: 1

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

nonn

a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - Thomas Ordowski, Jul 20 2012
Conjecture: a(n) < log(A014574(n))^2. - Thomas Ordowski, Jul 21 2012
Conjecture: All positive integers are represented in this sequence. This is verified up to 184, by searching up to prime indexes of ~128000000. The rate of filling-in the smallest remaining gap among the integers, and the growth in the maximum value found, both slow down considerably relative to a fixed quantity of twin prime incidences examined in each pass. The maximum value found was 237. - Richard R. Forberg, Jul 28 2016
All positive integers below 312 are in this sequence. - Charles R Greathouse IV, Aug 01 2016
From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal antirun of prime numbers > 3, where an antirun is an interval of positions at which consecutive terms differ by more than 2. These 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
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

First differences of A029707 and A155752 = A029707 - 1. M. F. Hasler, Jul 24 2012
Positions of first appearances are A373401, sorted A373402.
Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.
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. A025157, A038664, A071148,  A176246, A350842, A373406.

A027833 := proc(n)
    local plow,phigh ;
    phigh := A001359(n+1) ;
    plow := A001359(n) ;
    numtheory[pi](phigh)-numtheory[pi](plow) ;
end proc:
seq(A027833(n),n=1..100) ; # R. J. Mathar, Jan 20 2025

Differences[Flatten[Position[Differences[Prime[Range[500]]],2]]] (* Harvey P. Dale, Nov 17 2018 *)
Length/@Split[Select[Range[4,10000],PrimeQ[#]&],#1+2!=#2&]//Most (* Gus Wiseman, Jun 11 2024 *)

n=1; p=5; forprime(q=7,1e3, if(q-p==2, print1(n", "); n=1, n++); p=q) \\ Charles R Greathouse IV, Aug 01 2016

def A027833(n) :
   a = [ ]
   st = 2
   for i in (3..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i-st)
         st = i
   return(a)
A027833(496) # Jani Melik, May 15 2014

A373127 Length of the n-th maximal antirun of squarefree numbers differing by more than one.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The sum of this antirun is given by A373411.

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

			Row-lengths of:
   1
   2
   3  5
   6
   7 10
  11 13
  14
  15 17 19 21
  22
  23 26 29
  30
  31 33
  34
  35 37
  38
  39 41
  42
  43 46
  47 51 53 55 57

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

Positions of first appearances are A373128, sorted A373200.
Functional neighbors: A007674, A027833 (partial sums A029707), A120992, A373403, A373408, A373409, A373411.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A077643 counts squarefree numbers with n bits, sum A373123.
Cf. A049093, A049094, A061398, A077641, A112925, A112926, A174965, A350842, A372683, A373125, A373126, A373197, A373198.

Length/@Split[Select[Range[100],SquareFreeQ],#1+1!=#2&]

A373411 Sum of the n-th maximal antirun of squarefree numbers differing by more than one.

Original entry on oeis.org

1, 2, 8, 6, 17, 24, 14, 72, 22, 78, 30, 64, 34, 72, 38, 80, 42, 89, 263, 58, 120, 127, 66, 136, 70, 144, 151, 78, 161, 168, 86, 360, 94, 293, 102, 208, 106, 216, 110, 224, 114, 233, 241, 379, 130, 264, 271, 138, 280, 142, 288, 600, 312, 158, 648, 166, 510, 351

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The length of this antirun is given by A373127.

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

			Row-sums of:
   1
   2
   3  5
   6
   7 10
  11 13
  14
  15 17 19 21
  22
  23 26 29
  30
  31 33
  34
  35 37
  38
  39 41
  42
  43 46
  47 51 53 55 57

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

The partial sums are a subset of A173143.
Functional neighbors: A007674, A373127 (firsts A373128, sorted firsts A373200), A373404, A373405, A373408, A373412, A373413.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
Cf. A025157, A049093, A061398, A077641, A077643, A112925, A112926, A143658, A372683, A373123, A373197, A373198.

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

A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

Original entry on oeis.org

1, 5, 7, 12, 18, 28, 40, 53, 71, 109, 170, 190, 198, 207, 236, 303, 394, 457, 606, 774, 1069, 1100, 1225, 1881, 1930, 1952, 2247, 2281, 3140, 3368, 3451, 3493, 3713, 3862, 4595, 4685, 6625, 8063, 8121, 8783, 12359, 12650, 14471, 14979, 15901, 17129, 19155

Offset: 1

Gus Wiseman, Jun 15 2024

nonn

The unsorted version is A373669.

			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
So the a(n)-th runs begin:
   1
  14  15
  20  21  22
  33  34  35  36
  54  55  56  57  58

For nonsquarefree runs we have A373199 (if increasing), firsts of A053797.
For squarefree antiruns see A373200, unsorted A373128, firsts of A373127.
For composite runs we have A373400, unsorted A073051, firsts of A176246.
For prime antiruns we have A373402.
For runs of non-prime-powers:
- length A110969, firsts A373669, sorted A373670 (this sequence):
- min A373676
- max A373677
- sum A373678
For runs of prime-powers:
- length A174965
- min A373673
- max A373674
- sum A373675
A000961 lists the powers of primes (including 1).
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers, without 1 A024619.
Cf. A005381, A008864, A014963, A027833, A038664, A356068, A373401, A373574.

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

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]}]

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[[#]]]&]

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A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

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A373127 Length of the n-th maximal antirun of squarefree numbers differing by more than one.

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A373411 Sum of the n-th maximal antirun of squarefree numbers differing by more than one.

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A373670 Numbers k such that the k-th run-length A110969(k) of the sequence of non-prime-powers (A024619) is different from all prior run-lengths.

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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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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.

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