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A379402 Rectangular array, read by descending antidiagonals: the Type 2 runlength index array of A039701 (primes mod 3); see Comments.

%I A379402 #14 Mar 31 2025 07:00:35
%S A379402 1,2,9,3,11,15,4,16,18,54,5,21,23,58,91,6,32,36,102,110,205,7,37,39,
%T A379402 129,160,272,194,8,40,46,161,167,419,271,139,10,47,55,174,238,499,416,
%U A379402 260,86,12,56,73,245,273,597,496,359,257,357,13,67,96,274,292
%N A379402 Rectangular array, read by descending antidiagonals: the Type 2 runlength index array of A039701 (primes mod 3); see Comments.
%C A379402 We begin with a definition of Type 2 runlength array, V(s), of any sequence s for which all the runs referred to have finite length:
%C A379402 Suppose s is a sequence (finite or infinite), and define rows of V(s) as follows:
%C A379402 (row 0) = s
%C A379402 (row 1) = sequence of last terms of runs in (row 0); c(1) = complement of (row 1) in (row 0)
%C A379402 For n>=2,
%C A379402 (row n) = sequence of last terms of runs in c(n-1); c(n) = complement of (row n) in (row n-1),
%C A379402 where the process stops if and when c(n) is empty for some n.
%C A379402 ***
%C A379402 The corresponding Type 2 runlength index array,  The runlength index array, VI(s) is now contructed from V(s) in two steps:
%C A379402 (1) Let V*(s) be the array obtaining by repeating the construction of V(s) using (n,s(n)) in place of s(n).
%C A379402 (2) Then VI(s) results by retaining only n in V*.
%C A379402 Thus, loosely speaking, (row n) of VI(s) shows the indices in s of the numbers in (row n) of V(s).
%C A379402 The array VI(s) includes every positive integer exactly once.
%C A379402 ***
%C A379402 Regarding the present array, each row of U(s) splits a sequence of primes according to remainder modulo 3; e.g., in row 2, the remainders of primes in positions 9,11,16,21,32,37,40,47 are 2,1,2,1,2,1,2,1, respectively.
%C A379402 Conjecture: every column is eventually increasing.
%e A379402 Corner:
%e A379402       1    2    3    4      5     6     7     8    10    12    13    14
%e A379402       9   11   16   21     32    37    40    47    56    67    71    74
%e A379402      15   18   23   36     39    46    55    73    96    99   107   111
%e A379402      54   58  102  129    161   174   245   274   311   326   423   515
%e A379402      91  110  160  167    238   273   292   321   420   508   598   621
%e A379402     205  272  419  499    597   618   703   733   813   835   896   932
%e A379402     194  271  416  496    576   617   702   730   776   834   989  1128
%e A379402     139  260  359  489    699   713   771   831   988  1127  1173  1190
%e A379402      86  257  358  464    698   830   987  1124  1164  1185  1251  1298
%e A379402     357  461  697  829    942  1107  1412  1498  1717  2059  2138  2179
%e A379402     356  438  889  1062  1714  2046  2137  2176  2551  2820  2927  3270
%e A379402     291  437  882  1055  1711  2033  2550  2741  2926  3269  3699  3918
%e A379402 Starting with s = A039701, we have for U*(s):
%e A379402 (row 1) = ((1,1), (2,0), (3,2), (4,2), (5,2), (6,1), (7,2), (8,1), (10,2), ...)
%e A379402 c(1) = ((9,2), (11,1), (15,2), (16,2), (18,1), (21,1), (23,1), (32,2), ...)
%e A379402 (row 2) = ((9,2), (11,1), (16,2), (21,1), (36,1), ...)
%e A379402 c(2) = ((15,2), (37,1), ...)
%e A379402 (row 3) = ((15,2), (18,1), (23,2), ...)
%e A379402 so that UI(s) has
%e A379402 (row 1) = (1,2,3,4,5,6,7,8,10,12,13, ...)
%e A379402 (row 2) = (9,11,16.21,32, ...)
%e A379402 (row 3) = (15,18,23,...)
%t A379402 r[seq_] := seq[[Flatten[Position[Append[Differences[seq[[All, 1]]], 1], _?(# != 0 &)]], 2]];  (* Type 2 *)
%t A379402 row[0] = Mod[Prime[Range[4000]], 3];(* A039701 *)
%t A379402 row[0] = Transpose[{#, Range[Length[#]]}] &[row[0]];
%t A379402 k = 0; Quiet[While[Head[row[k]] === List, row[k + 1] = row[0][[r[
%t A379402      SortBy[Apply[Complement, Map[row[#] &, Range[0, k]]], #[[2]] &]]]]; k++]];
%t A379402 m = Map[Map[#[[2]] &, row[#]] &, Range[k - 1]];
%t A379402 p[n_] := Take[m[[n]], 12]
%t A379402 t = Table[p[n], {n, 1, 12}]
%t A379402 Grid[t]  (* array *)
%t A379402 w[n_, k_] := t[[n]][[k]];
%t A379402 Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten  (* sequence *)
%t A379402 (* _Peter J. C. Moses_, Dec 04 2024 *)
%Y A379402 Cf. A000040, A039701, A379046, A379401, A379403, A379404.
%K A379402 nonn,tabl
%O A379402 1,2
%A A379402 _Clark Kimberling_, Jan 15 2025