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A285090
Rectangular array by antidiagonals: the array formed by arranging the rows of A285089 so that the first column is strictly increasing.
1
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).
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
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Apr 13 2017
STATUS
approved