A389587 Increasing partition array based on the fractions A168265(n)/A389810(n); see Comments.

1, 3, 2, 7, 5, 4, 13, 6, 9, 8, 23, 12, 10, 15, 14, 35, 21, 11, 16, 25, 24, 51, 22, 20, 17, 26, 37, 36, 69, 34, 32, 18, 27, 38, 53, 52, 91, 50, 33, 19, 28, 39, 54, 71, 70, 119, 68, 49, 31, 29, 40, 55, 72, 93, 92, 149, 90, 67, 46, 30, 41, 56, 73, 94, 121, 120

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 fractions A168265(n)/A389810(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..66.

Example

The first 7 fractions with approximations are
  f(1) = 1/2 = 0.5000
  f(2) = 1/3 = 0.3333
  f(3) = 2/3 = 0.6667
  f(4) = 1/5 = 0.2000
  f(5) = 2/5 = 0.4000
  f(6) = 3/5 = 0.6000
  f(7) = 4/5 = 0.8000,
so (row 1) begins with 1,3,7; (row 2) begins with 2,5,6; and (row 3) begins with 4.
Corner:
     1    3    7   13   23   35   51   69   91  119
     2    5    6   12   21   22   34   50   68   90
     4    9   10   11   20   32   33   49   67   89
     8   15   16   17   18   19   31   46   47   48
    14   25   26   27   28   29   30   45   63   64
    24   37   38   39   40   41   42   43   44   62
    36   53   54   55   56   57   58   59   60   61
    52   71   72   73   74   75   76   77   78   79
    70   93   94   95   96   97   98   99  100  101

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[k/Prime[n], {n, 1, 30}, {k, 1, Prime[n] - 1}]];  A168265(n)/A389810(n)
(* 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]]

Crossrefs

Cf. A000027, A168265, A388853, A389586, A389810.

Sequence in context: A087468 A255975 A167267 * A097286 A278503 A283940

Adjacent sequences: A389584 A389585 A389586 * A389588 A389589 A389590

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 21 2025

Status

approved