A387594 Rectangular array (d(n,k)) read by descending antidiagonals: d(n,k) is the denominator array of the decreasing (2)-prime-fractions array, defined in Comments.

2, 3, 7, 5, 13, 31, 11, 17, 59, 71, 23, 19, 61, 101, 263, 29, 37, 79, 107, 311, 331, 41, 43, 103, 149, 389, 421, 569, 53, 47, 109, 211, 449, 433, 673, 631, 83, 67, 151, 257, 479, 499, 821, 977, 691, 89, 73, 163, 317, 571, 601, 839, 1091, 1279, 1487

Offset

1,1

Comments

Notation: Suppose that r>=1. If p is a prime, then p' denotes the least prime > r*p(i.e., in Mathematica code, p' = NextPrime[r*p]). Define d(1,1) = 2. and q(1,1) = d'(1,1)/d(1,1) . For k>=2, define d(1,k) = least prime p > d(1,k-1) such that p'/p < d'(1,k-1)/d (1,k-1), and define q(1,k) = p'/p . This finishes defining (row 1 of V(r)) = (d'(1,k)/d(1,k)), where k>=1. For n>=2, define d(n,1) = least prime p not in {d(h,k): h = 1..n-1, k >= 1}, and for k>=2 define d(n,k) = least prime p > d (n,k-1) such that p'/p < d'(n,k-1)/d (n,k-1), and define q(n,k) = p'/p. This finishes defining all rows of V(r). The denominator array of V(r) is the array (d(n, k): n>=1, k>=1).

Every prime occurs exactly once in the denominator array of V(2).

Links

Table of n, a(n) for n=1..55.

Example

Rows of the decreasing (2)-prime-fractions array, V(2):
  (row 1) = 5/2 > 7/3 > 11/5 > 23/11 > 47/23 > 59/29 > ...
  (row 2) = 17/7 > 29/13 > 37/17 > 41/19 > 79/37 > ...
  (row 3) = 67/31 > 127/59 > 127/61 > 163/79 > ...
Corner of the denominator array:
    2    3    5   11   23   29   41
    7   13   17   19   37   43   47
   31   59   61   79  103  109  151
   71  101  107  149  211  257  317
  263  311  389  449  479  571  577

Mathematica

lows := {First /@ #, Most[FoldList[Plus, 1, Length /@ #]]} &[
Split[Rest[FoldList[Min, +\[Infinity], #]]]] &;
unSortedComplement[list1_, list2_] :=
DeleteCases[list1, Apply[Alternatives, list2]];
r = 1; (* User; put your r>=1 here. *)
rh = {};
rStart = Map[NextPrime[r*#]/# &[Prime[#]] &, Range[150]];
AppendTo[rh, lows[rStart][[1]]]; While[Last[rh] =!= {},
AppendTo[rh, Reverse[#[[lows[Denominator[#]][[2]]]]] &[
Reverse[lows[unSortedComplement[rStart, Flatten[rh]]][[1]]]]]];
rh
Denominator[rh]
Grid[Most[Denominator[rh]], Frame -> All]
(* Peter J. C. Moses, Sep 01 2025 *)

Crossrefs

Cf. A387265, A387593.

Sequence in context: A192175 A294205 A059459 * A124440 A067363 A083188

Adjacent sequences: A387591 A387592 A387593 * A387595 A387596 A387597

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Sep 03 2025

Status

approved