cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Gus Wiseman, Jun 10 2024

Keywords

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.

Examples

			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

Links

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

Programs

  • Mathematica
    t=Length/@Split[Select[Range[4,10000],PrimeQ],#1+2!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A375737 Sum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.

Original entry on oeis.org

2, 8, 6, 17, 11, 12, 13, 14, 32, 18, 19, 20, 21, 22, 23, 78, 29, 30, 64, 34, 72, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 98, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 128, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 162, 83, 84, 85, 86, 87
Offset: 1

Views

Author

Gus Wiseman, Sep 10 2024

Keywords

Comments

Non-perfect-powers (A007916) are numbers with no proper integer roots.
An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.

Examples

			The initial anti-runs are the following, whose sums are a(n):
  (2)
  (3,5)
  (6)
  (7,10)
  (11)
  (12)
  (13)
  (14)
  (15,17)
  (18)
  (19)
  (20)
  (21)
  (22)
  (23)
  (24,26,28)

Crossrefs

For nonprime numbers we have A373404, runs A054265.
For squarefree numbers we have A373411, runs A373413.
For nonsquarefree numbers we have A373412, runs A373414.
For prime-powers we have A373576, runs A373675.
For non-prime-powers we have A373679, runs A373678.
For anti-runs of non-perfect-powers:
- length: A375736
- first: A375738
- last: A375739
- sum: A375737 (this)
For runs of non-perfect-powers:
- length: A375702
- first: A375703
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
Cf. A007674, A045542, A046933, A053797, A216765, A251092, A373403, A375714.

Programs

  • Mathematica
    radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Total/@Split[Select[Range[100],radQ],#1+1!=#2&]//Most

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

Views

Author

Gus Wiseman, Jun 10 2024

Keywords

Comments

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.

Examples

			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

Links

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.
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.

Programs

  • Mathematica
    t=Length/@Split[Select[Range[10000],SquareFreeQ],#1+1!=#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]
Previous Showing 11-13 of 13 results.

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