A389586 Increasing partition array based on the prime fractions A037126(n)/A389809(n); see Comments.

1, 6, 2, 15, 3, 4, 28, 10, 5, 7, 55, 20, 9, 8, 11, 91, 21, 14, 13, 12, 16, 153, 36, 26, 19, 18, 17, 22, 210, 66, 27, 33, 25, 24, 23, 29, 351, 78, 35, 34, 41, 32, 31, 30, 37, 406, 105, 45, 43, 42, 50, 40, 39, 38, 46, 561, 190, 90, 44, 52, 51, 60, 49, 48, 47

Offset

1,2

Comments

Suppose that S = (s(m)), for m >= 1, is a sequence of distinct real numbers that is dense in an open interval (a,b), such as the numbers (sin(n)), for n>=1, dense in (-1,1). The increasing partition array (p(n,k)) of the set N of positive integers is defined inductively as follows: p(1,1) = 1, and for k >= 2, p(1,k) = least m such that s(m) > s(p(1,k-1)). For n>=2, p(n,1) = least new m (that is, m is not p(h,k) for any h<=n-1 and k>=1), and for k>=2, p(n,k) = least new m such that s(m) > s(p(n,k-1)).

The decreasing partition array (p(n,k)) of N is defined as follows: p(1,1)=s(1), and for k>=2, p(1,k) = least new m such that s(m) < s(p(1,k-1)). For n>=2, p(n,1) = least new m, and for k>=2, p(n,k) = least new m such that s(m) < (p(n,k-1)). The decreasing partition array based on the prime fractions A037126(n)/A389809(n) is the natural number array, as in A000027 and A185787.

For a guide to related partition arrays, see A388853.

Links

Table of n, a(n) for n=1..65.

Example

The first 7 prime fractions with approximations are
  f(1) = 2/3 = 0.6667
  f(2) = 2/5 = 0.4000
  f(3) = 3/5 = 0.6000
  f(4) = 2/7 = 0.2857
  f(5) = 3/7 = 0.4286
  f(6) = 5/7 = 0.7143
  f(7) = 2/11 = 0.1818,
so (row 1) begins with 1,6; (row 2) begins with 2,3; (row 3) begins with 4,5; (row 4) begins with 7,8.
Corner:
    1    6   15   28   55   91  153  210  351  406
    2    3   10   20   21   36   66   78  105  190
    4    5    9   14   26   27   35   45   90  104
    7    8   13   19   33   34   43   44   54   65
   11   12   18   25   41   42   52   53   64   76
   16   17   24   32   50   51   62   63   75   88
   22   23   31   40   60   61   73   74   87  101
   29   30   39   49   71   72   85   86  100  115
   37   38   48   59   83   84   98   99  114  130
   46   47   58   70   96   97  112  113  129  146
   56   57   69   82  110  111  127  128  145  163
   67   68   81   95  125  126  143  144  162  181

Mathematica

highs := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[
    Split[Rest[FoldList[Max, -\[Infinity], #]]]] &;
lows := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[
    Split[Rest[FoldList[Min, +\[Infinity], #]]]] &;
seqS = Flatten[Table[Prime[k]/Prime[n], {n, 2, 50}, {k, 1, n - 1}]];
(* User: put your dense sequence S after seqS = *)
indices = Range[Length[seqS]];
arrI = {};   (*start accumulating increasing partition array*)
Until[Last[arrI] == {}, AppendTo[arrI, Flatten[Map[Position[seqS, #] &,
     highs[seqS[[Complement[indices, Flatten[arrI]]]]][[1]]]]]];
Grid[Take[arrI, 12]]
arrD = {};   (*start accumulating decreasing partition array*)
Until[Last[arrD] == {}, AppendTo[arrD, Flatten[Map[Position[seqS, #] &,
     lows[seqS[[Complement[indices, Flatten[arrD]]]]][[1]]]]]];
Grid[Take[arrD, 12]]
(* Peter J. C. Moses, Sep 04 2025 *)

Crossrefs

Cf. A000027, A037126, A388853, A389588 A389809.

Sequence in context: A080977 A095951 A116420 * A069268 A348060 A082155

Adjacent sequences: A389583 A389584 A389585 * A389587 A389588 A389589

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 21 2025

Status

approved