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

1,2

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).

Links

Table of n, a(n) for n=1..64.

Index entries for sequences that are permutations of the natural numbers

Example

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

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

Crossrefs

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

Sequence in context: A163990 A082156 A283941 * A114577 A285089 A191426

Adjacent sequences: A285087 A285088 A285089 * A285091 A285092 A285093

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Apr 13 2017

Status

approved