A360371 - OEIS

%I #41 Mar 30 2023 11:51:56

%S 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 34,22,23,27,30,40,52,51,44,28,29,33,35,48,65,68,66,56,31,36,42,45,64,

%U 78,85,88,84,62,37,38,54,50,72,91,102,110,112,93,74,41

%N 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 A permutation of the natural numbers.

%C 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 Rémy Sigrist, <a href="/A360371/b360371.txt">Table of n, a(n) for n = 1..10011</a> (first 141 rows flattened)

%H Samuel Harkness, <a href="/A360371/a360371.jpg">First 500000 terms</a>

%H Samuel Harkness, <a href="/A360371/a360371.m.txt">MATLAB program</a>

%H Rémy Sigrist, <a href="/A360371/a360371.gp.txt">PARI program</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e The start of the sequence as a triangular array read by rows:

%e    1;

%e    2,  3;

%e    4,  6,  5;

%e    7,  9, 10,  8;

%e   11, 12, 15, 16, 13;

%e   14, 18, 20, 24, 26, 17;

%e   19, 21, 25, 32, 39, 34, 22;

%e   23, 27, 30, 40, 52, 51, 44, 28;

%e   ...

%e Note that each column contains only multiples of the first number in the column.

%e 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 b:= proc() false end:

%p T:= proc(n, k) option remember; local j;

%p       if {n, k} = {1} then j:=1

%p     elif n=k then for j from T(n-1$2) while b(j) do od

%p     else for j from T(n-1, k) by T(k, k) while b(j) do od

%p       fi; b(j):=true; j

%p     end:

%p seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Mar 19 2023

%o (MATLAB) See Links section.

%Y Cf. A194982, A361251 (inverse).

%K nonn,tabl

%O 1,2

%A _Samuel Harkness_, Mar 17 2023