A389580 Increasing partition array based on the fractional parts of (n*e); see A388853.

1, 4, 2, 11, 5, 3, 18, 8, 6, 7, 25, 15, 9, 10, 14, 32, 22, 12, 13, 17, 21, 71, 29, 19, 16, 20, 24, 28, 536, 36, 26, 23, 27, 31, 35, 39, 1537, 43, 33, 30, 34, 38, 42, 46, 78, 2538, 50, 40, 37, 41, 45, 49, 53, 85, 110, 3539, 57, 47, 44, 48, 52, 56, 60, 92, 117

Offset

1,2

Links

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

Example

Fractional parts of e*n: 0.718..., 0.436..., 0.154..., 0.873..., 0.591..., 0.309..., 0.027..., etc., from which the first few terms of the increasing and decreasing partition arrays can be checked.
Corner:
    1   4  11  18   25   32   71  536
    2   5   8  15   22   29   36   43
    3   6   9  12   19   26   33   40
    7  10  13  16   23   30   37   44
   14  17  20  27   34   41   48   55
   21  24  31  38   45   52   59   66
   28  35  42  49   56   63   70   77
   39  46  53  60   67   74   81   88
   78  85  92  99  106  113  120  127

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 = Table[N[FractionalPart[E*n], 20], {n, 1, 1000}];
(* 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. A388853, A389581.

Sequence in context: A336914 A390605 A389576 * A091484 A163544 A191728

Adjacent sequences: A389577 A389578 A389579 * A389581 A389582 A389583

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 15 2025

Status

approved