A285090 Rectangular array by antidiagonals: the array formed by arranging the rows of A285089 so that the first column is strictly increasing.

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 21, 7, 36, 30, 24, 32, 27, 10, 49, 42, 35, 45, 55, 18, 11, 64, 56, 48, 60, 91, 28, 39, 13, 81, 72, 63, 77, 112, 40, 75, 85, 14, 100, 90, 80, 96, 135, 54, 119, 133, 50, 17, 121, 110, 99, 117, 160, 70, 171, 189, 66

Offset: 1

Author

Clark Kimberling, Apr 13 2017

Keywords

• nonn
• tabl
• easy

Comments

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the natural numbers, A000027. Every prime (A000040) occurs in column 1. For each row, there is a nonnegative integer h such that all but finitely many initial entries are of the form k*(k+h).

Examples

			Northwest corner:
1   4   9    16   25   36   49   64   81   10
2   6   12   20   30   42   56   72   90   110
3   8   15   24   35   48   63   80   99   120
5   21  32   45   60   77   96   117  140  165
7   27  55   91   112  135  160  187  216  247
10  18  28   40   54   70   88   108  130  154
11  39  75   119  171  200  231  264  299  375
13  85  133  189  253  325  364  405  448  493
		

Links

• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000027, A000040, A000290, A095833, A163280, A285089.

Programs

Mathematica

d[n_] := Divisors[n]; k[n_] := Length[d[n]]; x[n_, i_] := d[n][[i]];
a[n_] := If[OddQ[k[n]], 0, x[n, k[n]/2 + 1] - x[n, k[n]/2]]
t = Table[a[j], {j, 1, 30000}];
r[n_] := Flatten[Position[t, n]]; v[n_, k_] := r[n][[k]];
w = Table[v[n, k], {n, 0, 20}, {k, 1, 20}];
y = SortBy[w, First]; v[n_, k_] := y[[n, k]];
w = TableForm[Table[v[n, k], {n, 1, 10}, {k, 1, 10}]]
Table[v[n + 1 - k, k], {n, 1, 15}, {k, n, 1, -1}] // Flatten