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A360371 Triangle read by rows: lexicographically earliest sequence of distinct positive integers such that each column contains only multiples of the first number in that column. See example.

%I A360371 #41 Mar 30 2023 11:51:56
%S A360371 1,2,3,4,6,5,7,9,10,8,11,12,15,16,13,14,18,20,24,26,17,19,21,25,32,39,
%T A360371 34,22,23,27,30,40,52,51,44,28,29,33,35,48,65,68,66,56,31,36,42,45,64,
%U A360371 78,85,88,84,62,37,38,54,50,72,91,102,110,112,93,74,41
%N A360371 Triangle read by rows: lexicographically earliest sequence of distinct positive integers such that each column contains only multiples of the first number in that column. See example.
%C A360371 A permutation of the natural numbers.
%C A360371 Among the first number of columns, are there more primes or composites? Of the first 500 columns, 296 are prime, 203 are composite (first column begins with 1).
%H A360371 Rémy Sigrist, <a href="/A360371/b360371.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows flattened)
%H A360371 Samuel Harkness, <a href="/A360371/a360371.jpg">First 500000 terms</a>
%H A360371 Samuel Harkness, <a href="/A360371/a360371.m.txt">MATLAB program</a>
%H A360371 Rémy Sigrist, <a href="/A360371/a360371.gp.txt">PARI program</a>
%H A360371 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A360371 The start of the sequence as a triangular array read by rows:
%e A360371    1;
%e A360371    2,  3;
%e A360371    4,  6,  5;
%e A360371    7,  9, 10,  8;
%e A360371   11, 12, 15, 16, 13;
%e A360371   14, 18, 20, 24, 26, 17;
%e A360371   19, 21, 25, 32, 39, 34, 22;
%e A360371   23, 27, 30, 40, 52, 51, 44, 28;
%e A360371   ...
%e A360371 Note that each column contains only multiples of the first number in the column.
%e A360371 For a(17), note that we are in the second column, so a(17) must be a positive multiple of 3. No numbers can be repeated, and we see that {3, 6, 9, 12, 15} have already been used, and 18 is the smallest unused positive multiple of 3. Therefore, a(17) = 18.
%p A360371 b:= proc() false end:
%p A360371 T:= proc(n, k) option remember; local j;
%p A360371       if {n, k} = {1} then j:=1
%p A360371     elif n=k then for j from T(n-1$2) while b(j) do od
%p A360371     else for j from T(n-1, k) by T(k, k) while b(j) do od
%p A360371       fi; b(j):=true; j
%p A360371     end:
%p A360371 seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Mar 19 2023
%o A360371 (MATLAB) See Links section.
%Y A360371 Cf. A194982, A361251 (inverse).
%K A360371 nonn,tabl
%O A360371 1,2
%A A360371 _Samuel Harkness_, Mar 17 2023