A285089 Rectangular array by antidiagonals: row n is the ordered sequence of numbers k that minimize |d(n+1-k) - d(k)|, where d(i) are the divisors of n.

1, 4, 2, 9, 6, 3, 16, 12, 8, 10, 25, 20, 15, 18, 5, 36, 30, 24, 28, 21, 14, 49, 42, 35, 40, 32, 50, 7, 64, 56, 48, 54, 45, 66, 27, 44, 81, 72, 63, 70, 60, 84, 55, 78, 33, 100, 90, 80, 88, 77, 104, 91, 98, 65, 22, 121, 110, 99, 108, 96, 126, 112, 170, 105, 52

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.

Row 1: A000290 (squares)

Row 2: A002378 (oblong numbers)

Row 3: A005563

Row 4: A028552 (for n>=2)

Examples

			Taking n = 12, the divisors are 1,2,3,4,6,12, so that for k=1..6, the numbers d(n+1-k) - d(k) are 12-1, 6-2, 4-3, 3-4, 2-6, 1-12.  Thus, the number k that minimizes |d(n+1-k) - d(k)| is 1, so that 12 appears in row 1 (with the top row as row 0), consisting of numbers for which the minimal value is 1.
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
  10  18  28  40   54   70   88   108  130  154
  5   21  32  45   60   77   96   117  140  165
  14  50  66  84   104  126  160  176  204  234
  7   27  55  91   112  135  160  187  216  247
  44  78  98  170  198  228  260  294  330  368
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000027, A000040, A000290, A095833, A163280, A285090.

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, 10}, {k, 1, 10}];
TableForm[w] (* A285089, array *)
Table[v[n - k, k], {n, 0, 60}, {k, n, 1, -1}] // Flatten (* A285089, sequence *)

Formula

row 1: k^2 for k>=1

row 2: k*(k+1) for k>=1

row 3: k*(k+2) for k>=3

row 4: k*(k+3) for k>=2

row 5: k*(k+4) for k>=3

row 6: k*(k+5) for k>=5

row 7: k*(k+6) for k>=7