A389586 - OEIS

A389586

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

%I #12 Oct 30 2025 18:38:42

%S 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,

%T 18,17,22,210,66,27,33,25,24,23,29,351,78,35,34,41,32,31,30,37,406,

%U 105,45,43,42,50,40,39,38,46,561,190,90,44,52,51,60,49,48,47

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

%C 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)).

%C 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.

%C For a guide to related partition arrays, see A388853.

%e The first 7 prime fractions with approximations are

%e f(1) = 2/3 = 0.6667

%e f(2) = 2/5 = 0.4000

%e f(3) = 3/5 = 0.6000

%e f(4) = 2/7 = 0.2857

%e f(5) = 3/7 = 0.4286

%e f(6) = 5/7 = 0.7143

%e f(7) = 2/11 = 0.1818,

%e 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.

%e Corner:

%e 1 6 15 28 55 91 153 210 351 406

%e 2 3 10 20 21 36 66 78 105 190

%e 4 5 9 14 26 27 35 45 90 104

%e 7 8 13 19 33 34 43 44 54 65

%e 11 12 18 25 41 42 52 53 64 76

%e 16 17 24 32 50 51 62 63 75 88

%e 22 23 31 40 60 61 73 74 87 101

%e 29 30 39 49 71 72 85 86 100 115

%e 37 38 48 59 83 84 98 99 114 130

%e 46 47 58 70 96 97 112 113 129 146

%e 56 57 69 82 110 111 127 128 145 163

%e 67 68 81 95 125 126 143 144 162 181

%t highs := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[

%t Split[Rest[FoldList[Max, -\[Infinity], #]]]] &;

%t lows := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[

%t Split[Rest[FoldList[Min, +\[Infinity], #]]]] &;

%t seqS = Flatten[Table[Prime[k]/Prime[n], {n, 2, 50}, {k, 1, n - 1}]];

%t (* User: put your dense sequence S after seqS = *)

%t indices = Range[Length[seqS]];

%t arrI = {}; (*start accumulating increasing partition array*)

%t Until[Last[arrI] == {}, AppendTo[arrI, Flatten[Map[Position[seqS, #] &,

%t highs[seqS[[Complement[indices, Flatten[arrI]]]]][[1]]]]]];

%t Grid[Take[arrI, 12]]

%t arrD = {}; (*start accumulating decreasing partition array*)

%t Until[Last[arrD] == {}, AppendTo[arrD, Flatten[Map[Position[seqS, #] &,

%t lows[seqS[[Complement[indices, Flatten[arrD]]]]][[1]]]]]];

%t Grid[Take[arrD, 12]]

%t (* _Peter J. C. Moses_, Sep 04 2025 *)

%Y Cf. A000027, A037126, A388853, A389588 A389809.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Oct 21 2025