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A386755 Triangle read by rows, where row terms are filled depending on divisibility of n. See comments.

%I A386755 #24 Aug 21 2025 16:48:52
%S A386755 1,0,1,0,2,1,0,2,0,1,0,0,3,2,1,0,0,3,2,0,1,0,0,3,0,0,2,1,0,0,3,0,0,2,
%T A386755 0,1,0,0,0,4,0,3,0,2,1,0,0,0,4,0,3,0,2,0,1,0,0,0,0,5,0,0,4,3,2,1,0,0,
%U A386755 0,0,5,0,0,4,3,2,0,1,0,0,0,0,5,0,0,4,3,0,0,2,1
%N A386755 Triangle read by rows, where row terms are filled depending on divisibility of n. See comments.
%C A386755 The construction of the array is as follows:
%C A386755 With divisibility testing steps of the process, we find the next strictly smaller multiple of k and enter that k in the row, one after the other.
%C A386755 In this process of row by row, we start at column number n that is the same as the row number. We always start with k=1 and then seek out those greatest m's that are lesser than the previous ones and which are divisible by the incremental k's, until a k exceeds the next m to be tested for divisibility. We write the divisors k inside the rows of the array, but leave 0 where k is not a divisor of n.
%C A386755 Notable properties of the array: Each column contains the proper divisors of n as well as n itself, with multiplicity.
%C A386755 The construction of row n is independent of any other row.
%C A386755 Another related sequence, A007952, is the row numbers where the first k=n's appear.
%C A386755 To construct row n we start by placing d=1 in column n. If we just placed d-1 in column j, then we next place d in the largest column k < j such that d | k. If there is no such column the process ends with d-1 as the largest number in the row. All unused columns are set to 0.
%e A386755 The triangle begins:
%e A386755   1;
%e A386755   0, 1;
%e A386755   0, 2, 1;
%e A386755   0, 2, 0, 1;
%e A386755   0, 0, 3, 2, 1;
%e A386755   0, 0, 3, 2, 0, 1;
%e A386755   0, 0, 3, 0, 0, 2, 1;
%e A386755   0, 0, 3, 0, 0, 2, 0, 1;
%e A386755   0, 0, 0, 4, 0, 3, 0, 2, 1;
%e A386755   0, 0, 0, 4, 0, 3, 0, 2, 0, 1;
%e A386755   ...
%e A386755 .
%e A386755 An example for the step by step construction of a particular row, let it be row n=20: We start with k=1 at column 20, and find that k=1 divides c=20. So we enter k=1 into the array in that column. Next, let now k=2, and we look for the greatest c that is less than 20, and which is divisible by k=2. That c is 18 in column 18, so we enter 2 in that column. We increase k by 1 to k=3, and similarly seek out the greatest c again that is less than 18, and which is divisible by 3. This number is 15, and so we enter 3 in column 15. And so on, we test divisibility with k=4, k=5, and k=6 to find that these k's fit under c=12, c=10 and c=6, respectively.
%e A386755   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
%e A386755   0  0  0  0  0  6  0  0  0  5  0  4  0  0  3  0  0  2  0  1
%o A386755 (PARI) row(n) = my(v=vector(n), m=n); for(k=1, n, my(keepm = m); while(m%k, m--); if (m == 0, keepm=m, v[m] = k; m--);); v; \\ _Michel Marcus_, Aug 01 2025
%Y A386755 Cf. A007952, A386520.
%K A386755 nonn,tabl,changed
%O A386755 1,5
%A A386755 _Tamas Sandor Nagy_, Aug 01 2025