A387266 Rectangular array (d(n,k)) read by falling antidiagonals: d(n,k) is the denominator array of q(n,k) in the array V(1), defined in Comments.

2, 5, 3, 11, 7, 23, 17, 13, 47, 89, 29, 19, 53, 139, 113, 41, 31, 61, 181, 199, 467, 59, 37, 73, 241, 211, 619, 523, 71, 43, 83, 283, 293, 661, 839, 887, 101, 67, 131, 337, 317, 773, 863, 1069, 1129, 107, 79, 151, 359, 409, 797, 1259, 1637, 1669, 1327

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 odd prime occurs exactly once in the denominator array of V(1), and every prime occurs exactly once in the corresponding array of numerators.

Links

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

Example

Rows of the decreasing (1)-prime-fractions array V(1):
  (row 1) = 3/2 > 7/5 > 13/11 > 19/17 >  31/29 > ...
  (row 2) = 5/3 > 11/7 > 17/13 > 23/19 > 37/31 > ...
  (row 3) = 29/23 > 53/47 > 59/53 > 67/61 > 79/73 > ...
Corner of the denominator array:
    2    5   11   17   29   41   59
    3    7   13   19   31   43   67
   23   47   53   61   73   83  131
   89  139  181  241  283  337  359
  113  199  211  293  317  409  479

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. A000040, A387265.

Sequence in context: A229608 A185061 A129198 * A122442 A225258 A162613

Adjacent sequences: A387263 A387264 A387265 * A387267 A387268 A387269

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Aug 25 2025

Status

approved