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 A385000 #62 Jul 18 2025 00:28:20 %S A385000 1,1,2,1,0,3,1,0,0,4,1,0,2,0,5,1,0,0,0,0,6,1,0,0,0,0,0,7,1,0,0,2,3,0, %T A385000 0,8,1,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,10,1,0,0,0,2,0,4,0,0,0,11,1, %U A385000 0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,3,0,0,0,0,0,13,1,0,0,0,0,2,0,0,5,0,0,0,0,14 %N A385000 Square array read by upward antidiagonals: A(n,k) = 0 except for A(d*(m-1),m*(d-1)) = d, with n >= 0, k >= 0, d >= 1, m >= 1. %C A385000 Given a number d whose position is (n,k) so the next number d is in the position (n+d,k+d-1). In other words: we can find the next number d by walking d steps south and then walking d-1 steps east. Hence the Manhattan distance between two nearest numbers d is 2*d - 1. %C A385000 By definition the m-th number d is in the position (d*(m-1),m*(d-1)) thus all the numbers d are on the same straight line. %C A385000 The positive terms in the row n >= 1 are the divisors of n in increasing order. Hence the number of positive terms in row n is A000005(n). %C A385000 The positive terms in the column k >= 1 are 1 plus the divisors of k in decreasing order. Hence the number of positive terms in the column k is A000005(k). %C A385000 In the row n >= 1, two conjugate divisors of n are equidistant from the position (n,n-1). That position is in the same column that contains to the number n in the row 0. %C A385000 In the column k >= 1, two conjugate (1 plus divisor)'s of k are equidistant from the position (k,k+1). %C A385000 On the other hand we can find the divisors of n on a curve which starts at A(n-1,0) = 1 and ends at A(0,n-1) = n. Hence the number of positive terms in the curve (n-1) is A000005(n). Here that curve is called "curve of divisors of n". %C A385000 Note that every divisor of n less to n on the curve is also a divisor of a number less to n in a row. Hence every divisor of n in a row is also a divisor on the curve of divisors of a number greater than n. %C A385000 The position of two conjugate divisors of n on the curve is orthogonal and equidistant to the main diagonal of the array. %C A385000 Curves never touch each other. See the curve and the row both with the divisors of 12 in the Example section. %C A385000 Drawing all the curves the 2D structure is compatible with a 3D model where in orthogonal position is the arc diagram of A000005. See Links section. %C A385000 The main diagonal is an irregular triangle read by rows in which row r lists r together with 2*r - 1 zeros, r >= 1. %H A385000 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv01.jpg">Illustration of initial terms of A000005 (arc diagram)</a> %e A385000 The corner 9 X 9 of the square array is as shown below: %e A385000 . %e A385000 \k 0 1 2 3 4 5 6 7 8 %e A385000 n\ _ _ _ _ _ _ _ _ _ _ %e A385000 | %e A385000 0 | 1 2 3 4 5 6 7 8 9 %e A385000 1 | 1 0 0 0 0 0 0 0 0 %e A385000 2 | 1 0 2 0 0 0 0 0 0 %e A385000 3 | 1 0 0 0 3 0 0 0 0 %e A385000 4 | 1 0 0 2 0 0 4 0 0 %e A385000 5 | 1 0 0 0 0 0 0 0 5 %e A385000 6 | 1 0 0 0 2 0 3 0 0 %e A385000 7 | 1 0 0 0 0 0 0 0 0 %e A385000 8 | 1 0 0 0 0 2 0 0 0 %e A385000 ... %e A385000 . %e A385000 The corner 25 X 25 of the square array without the zeros is as shown below: %e A385000 . %e A385000 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 %e A385000 \k 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 %e A385000 n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A385000 0 | 1 2 3 4 5 6 7 8 9 ... %e A385000 1 | 1 %e A385000 2 | 1 2 %e A385000 3 | 1 3 %e A385000 4 | 1 2 4 %e A385000 5 | 1 5 %e A385000 6 | 1 2 3 6 %e A385000 7 | 1 7 %e A385000 8 | 1 2 4 8 %e A385000 9 | 1 3 9 %e A385000 10 | 1 2 5 10 %e A385000 11 | 1 11 %e A385000 12 | 1 2 3 4 6 12 %e A385000 13 | 1 13 %e A385000 14 | 1 2 7 %e A385000 15 | 1 3 5 %e A385000 16 | 1 2 4 8 %e A385000 17 | 1 %e A385000 18 | 1 2 3 6 9 %e A385000 19 | 1 %e A385000 20 | 1 2 4 5 %e A385000 21 | 1 3 7 %e A385000 22 | 1 2 %e A385000 23 | 1 %e A385000 24 | 1 2 3 4 %e A385000 ... %e A385000 . %e A385000 The corner 25 X 25 of the square array without the zeros with the row and the curve of the divisors of 12 is as shown below: %e A385000 . %e A385000 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 %e A385000 \k 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 %e A385000 n\ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A385000 | %e A385000 0 | 12 %e A385000 1 | %e A385000 2 | %e A385000 3 | %e A385000 4 | %e A385000 5 | %e A385000 6 | 6 %e A385000 7 | %e A385000 8 | 4 %e A385000 9 | 3 %e A385000 10 | 2 %e A385000 11 | 1 %e A385000 12 | 1 2 3 4 6 12 %e A385000 ... %e A385000 . %e A385000 The position of the conjugate divisors of 12 on the curve is symmetric respect to the main diagonal of the square array. %e A385000 The position of the conjugate divisors of 12 in row 12 is symmetric respect the position (12,11). That position is in the same column that contains to the number 12 in the row 0. %Y A385000 Row sums give A000203, n >= 1. %Y A385000 Column sums give A007503 = A000203 + A000005, k >= 1. %Y A385000 Row 0 gives A000027. %Y A385000 Column 0 gives A000012. %Y A385000 Cf. A027750, A056538, A208460. %K A385000 nonn,tabl %O A385000 0,3 %A A385000 _Omar E. Pol_, Jun 16 2025