This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379401 #10 Mar 31 2025 07:00:16
%S A379401 1,2,10,3,12,17,4,16,22,56,5,19,33,75,57,6,24,38,97,134,98,7,37,41,
%T A379401 115,165,274,109,8,40,48,162,181,299,275,166,9,47,55,180,220,466,318,
%U A379401 276,241,11,52,68,201,273,554,467,363,279,256,13,59,92,264,294
%N A379401 Rectangular array, read by descending antidiagonals: the Type 1 runlength index array of A039701 (primes mod 3); see Comments.
%C A379401 We begin with a definition of Type 1 runlength array, U(s), of a sequence s:
%C A379401 Suppose s is a sequence (finite or infinite), and define rows of U(s) as follows:
%C A379401 (row 0) = s
%C A379401 (row 1) = sequence of 1st terms of runs in (row 0); c(1) = complement of (row 1) in (row 0)
%C A379401 For n>=2,
%C A379401 (row n) = sequence of 1st terms of runs in c(n-1); c(n) = complement of (row n) in (row n-1),
%C A379401 where the process stops if and when c(n) is empty for some n.
%C A379401 ***
%C A379401 The corresponding Type 1 runlength index array, UI(s) is now contructed from U(s) in two steps:
%C A379401 (1) Let U*(s) be the array obtaining by repeating the construction of U(s) using (n,s(n)) in place of s(n).
%C A379401 (2) Then UI(s) results by retaining only n in U*.
%C A379401 Thus, loosely speaking, (row n) of UI(s) shows the indices in s of the numbers in (row n) of U(s).
%C A379401 The array UI(s) includes every positive integer exactly once.
%C A379401 ***
%C A379401 Regarding the present array, each row of U(s) splits an increasing sequence of primes according to remainder modulo 3; e.g., in row 2, the remainders of primes in positions 10,12,16,19,24,37 are 2,1,2,1,2,1,2,1, respectively.
%e A379401 Corner:
%e A379401 1 2 3 4 5 6 7 8 9 11 13 14
%e A379401 10 12 16 19 24 37 40 47 52 59 72 74
%e A379401 17 22 33 38 41 48 55 68 92 101 104 112
%e A379401 56 75 97 115 162 180 201 264 293 328 359 440
%e A379401 57 134 165 181 220 273 294 341 360 451 545 623
%e A379401 98 274 299 466 554 624 661 742 786 836 898 941
%e A379401 109 275 318 467 555 631 704 749 823 839 903 1046
%e A379401 166 276 363 500 600 758 824 856 912 1059 1176 1212
%e A379401 241 279 364 505 601 861 913 1076 1177 1229 1258 1368
%e A379401 256 510 608 866 964 1077 1180 1533 1645 2006 2156 2215
%e A379401 421 521 709 1088 1181 2007 2163 2248 2551 2690 2919 3138
%e A379401 424 522 710 1089 1184 2008 2174 2785 2920 3141 3466 3938
%e A379401 Starting with s = A039701, we have for U*(s):
%e A379401 (row 1) = ((1,1), (2,0), (3,2), (4,2), (5,2), (6,1), (7,2), (8,1), (9,2), ...)
%e A379401 c(1) = ((10,2), (12,1), (16,2), (17,2), (14,1), (17,1), (19,1), (22,1), (24,2), ...)
%e A379401 (row 2) = ((10,2), (12,1), (16,2), (19,1), (24,2), (23,1), (27,2), (29,1), (36,2), ...)
%e A379401 c(2) = ((17,2), (22,1), (33,2), ...)
%e A379401 (row 3) = ((17,2), (22,1), ...)
%e A379401 so that UI(s) has
%e A379401 (row 1) = (1,2,3,4,5,6,7,8,9,11,13, ...)
%e A379401 (row 2) = (10,12,16.19,24, ...)
%e A379401 (row 3) = (17,22,33,...)
%t A379401 r[seq_] := seq[[Flatten[Position[Prepend[Differences[seq[[All, 1]]], 1], _?(# != 0 &)]], 2]]; (* Type 1 *)
%t A379401 row[0] = Mod[Prime[Range[4000]], 3];(* A039701 *)
%t A379401 row[0] = Transpose[{#, Range[Length[#]]}] &[row[0]];
%t A379401 k = 0; Quiet[While[Head[row[k]] === List, row[k + 1] = row[0][[r[
%t A379401 SortBy[Apply[Complement, Map[row[#] &, Range[0, k]]], #[[2]] &]]]]; k++]];
%t A379401 m = Map[Map[#[[2]] &, row[#]] &, Range[k - 1]];
%t A379401 p[n_] := Take[m[[n]], 12]
%t A379401 t = Table[p[n], {n, 1, 12}]
%t A379401 Grid[t] (* array *)
%t A379401 w[n_, k_] := t[[n]][[k]];
%t A379401 Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* sequence *)
%t A379401 (* _Peter J. C. Moses_, Dec 04 2024 *)
%Y A379401 Cf. A000040, A039701, A379046, A379402, A379403, A379404.
%K A379401 nonn,tabl
%O A379401 1,2
%A A379401 _Clark Kimberling_, Jan 15 2025